The responsibility of parents is then to make their offspring Kṛṣṇa conscious.
(Bhagavad-Gita-----7:11-----purport).

A child is completely cared for by his parents, and thus his position is secure.

(Bhagavad-Gita-----12:6-7-----purport).

Parents are always tolerant to children for all sorts of mischievous acts.
(Srimad Bhagavatam-----1:12:22-----purport).

A son is indebted to his parents in so many ways, and it is the duty of the son to serve his parents, however great the son may be.
(Srimad Bhagavatam-----3:2:17-----purport).

Just as children are taken care of by the parents, women as a class should be taken care of by the father, husband or grown-up son. Those who are helpless must be taken care of by their respective guardians, otherwise the guardians will be subjected to the punishment of Yamarāja, who is appointed by the Lord to supervise the activities of sinful living creatures.
(Srimad Bhagavatam-----3:6:10-----purport).

The mental condition of a child depends upon the mental status of his parents at the time he is conceived. According to the Vedic system, therefore, the garbhādhāna-saṁskāra, or the ceremony for giving birth to a child, is observed. Before begetting a child, one has to sanctify his perplexed mind. When the parents engage their minds in the lotus feet of the Lord and in such a state the child is born, naturally good devotee children come; when the society is full of such good population, there is no trouble from demoniac mentalities.
(Srimad Bhagavatam-----3:20:28-----purport).

The selection of a good husband for a good girl was always entrusted to the parents. Here it is clearly stated that Manu and his wife were coming to see Kardama Muni to offer their daughter because the daughter was well qualified and the parents were searching out a similarly qualified man. This is the duty of parents. Girls are never thrown into the public street to search out their husband, for when girls are grown up and are searching after a boy, they forget to consider whether the boy they select is actually suitable for them. Out of the urge of sex desire, a girl may accept anyone, but if the husband is chosen by the parents, they can consider who is to be selected and who is not. According to the Vedic system, therefore, the girl is given over to a suitable boy by the parents; she is never allowed to select her own husband independently.
(Srimad Bhagavatam-----3:21:27-----purport).

Unless one can train a child for liberation in that life, there is no need to marry or produce children. If human society produces children like cats and dogs for the disturbance of social order, then the world becomes hellish, as it has in this age of Kali. In this age, neither parents nor their children are trained; both are animalistic and simply eat, sleep, mate, defend, and gratify their senses. This disorder in social life cannot bring peace to human society.
(Srimad Bhagavatam-----3:22:19-----purport).

By bodily union of the husband and wife their qualities are expanded: children born of good parents are expansions of the parents' personal qualifications.
(Srimad Bhagavatam-----3:23:10-----purport).

The parents used to astrologically determine the character and tastes of the boy and girl, and when they corresponded, the match was selected: "This girl and this boy are just suitable, and they should be married." Other considerations were less important. The same system was also advised in the beginning of the creation by Brahmā: "Your daughters should be handed over to the ṛṣis according to taste and character."
(Srimad Bhagavatam-----3:24:15-----purport).

To take care of the child is the primary duty of the parents because if such care is taken, society will be filled with good population to maintain the peace and prosperity of the society, country and human race.
(Srimad Bhagavatam-----3:31:5-----purport).

In childhood a woman must be under the protection of the parents, in youth she must be under the protection of the husband, and in old age she must be under the protection of the grown children.
(Srimad Bhagavatam-----3:33:19-----purport).

It is the duty of parents to hand over their daughters to suitable persons just befitting their family tradition in cleanliness, gentle behavior, wealth, social position, etc. 
(Srimad Bhagavatam-----4:2:16-----purport).

They are very carefully protected by their parents while unmarried, after marriage they are protected by their young husbands, and when elderly they are protected by their children. When thus protected, women as a class remain an always auspicious source of energy to man.
(Srimad Bhagavatam-----4:21:4-----purport).

The mental state of the father and mother before sex will certainly affect the mentality of the child to be begotten. A child who is begotten out of lust may not turn out as the parents desire. As stated in the śāstras, yathā yonir yathā bījam. Yathā yoniḥ indicates the mother, and yathā bījam indicates the father. If the mental state of the parents is prepared before they have sex, the child which they will beget will certainly reflect their mental condition.
(Srimad Bhagavatam-----4:22:53-----purport).

After the period of brahmacarya, a man accepts a householder's life, and the woman is also taught by her parents to be a chaste wife. Thus when a girl and boy are united, both are trained for a life dedicated to a higher purpose. The boy is trained to execute his duty in accordance with the higher purpose of life, and the girl is trained to follow him.
(Srimad Bhagavatam-----4:23:20-----purport).

The parent, after his death, cannot help his child. A father may leave some estate for his children's immediate help, but he should not be overly absorbed in thoughts of how his family will survive after his death. This is the disease of the conditioned soul. Not only does he commit sinful activities for his own sense gratification, but he accumulates great wealth to leave behind so that his children may also gorgeously arrange for sense gratification.
(Srimad Bhagavatam-----4:28:22-----purport).

Śrīla Viśvanātha Cakravartī Ṭhākura compares one's family to high mountains. Becoming happy in their association is like a hungry man's endeavoring to climb a mountain full of thorns. Almost 99.9 percent of the population is unhappy in family life, despite all the attempts being made to satisfy the family members. In the Western countries, due to the dissatisfaction of the family members, there is actually no family life. There are many cases of divorce, and out of dissatisfaction, the children leave the protection of their parents. Especially in this age of Kali, family life is being reduced. Everyone is becoming self-centered because that is the law of nature. Even if one has sufficient money to maintain a family, the situation is such that no one is happy in family life.
(Srimad Bhagavatam-----5:13:8-----purport).

It is customary in Hindu society for parents to give their children names like Kṛṣṇadāsa, Govinda dāsa, Nārāyaṇa dāsa and Vṛndāvana dāsa. Thus they chant the names Kṛṣṇa, Govinda, Nārāyaṇa and Vṛndāvana and get the chance to be purified.
(Srimad Bhagavatam-----6:1:25-----purport).

This animosity between parents and the Kṛṣṇa consciousness movement has existed since time immemorial. Even Nārada Muni was condemned, not to speak of others. Nevertheless, Nārada Muni never gives up his mission. To deliver as many fallen souls as possible, he continues playing his musical instrument and vibrating the transcendental sound Hare Kṛṣṇa, Hare Kṛṣṇa, Kṛṣṇa Kṛṣṇa, Hare Hare/ Hare Rāma, Hare Rāma, Rāma Rāma, Hare Hare.
(Srimad Bhagavatam-----6:5:23-----purport).

The real fact is that if the Supreme Personality of Godhead protects one, even though one be motherless and fatherless, one can be maintained by the good will of the Lord. Otherwise, if the Supreme Lord does not give one protection, one must suffer in spite of the presence of his father and mother. Another example is that sometimes a patient dies in spite of a good physician and good medicine. Thus without the protection of the Lord one cannot live, with or without parents.
(Srimad Bhagavatam-----7:2:55-----purport).

Despite the presence of a father and mother, a child cannot be protected from accidental death, disease and various other miseries. No one can help, including theparents. Ultimately the shelter is the Lord, and one who takes shelter of the Lord is protected. This is guaranteed.
(Srimad Bhagavatam-----7:9:19-----purport).

According to the Vedic system, the parents would consider the horoscopes of the boy and girl who were to be married. If according to astrological calculations the boy and girl were compatible in every respect, the match was called yoṭaka and the marriage would be accepted. Even fifty years ago, this system was current in Hindu society. Regardless of the affluence of the boy or the personal beauty of the girl, without this astrological compatibility the marriage would not take place.
(Srimad Bhagavatam-----9:18:23-----purport).

Generally it is understood that the protectors for a child are his parents, but this is not actually the fact. The real protector is the Supreme Personality of Godhead.
taptasya tat-pratividhir ya ihāñjaseṣṭas
tāvad vibho tanu-bhṛtāṁ tvad-upekṣitānām
(S.B.7:9:19)
If neglected by the Supreme Personality of Godhead, a child, despite the presence of his parents, will suffer, and a diseased person, despite all medical help, will die. In this material world, where there is a struggle for existence, men have invented many means for protection, but these are useless if the Supreme Personality of Godhead rejects them. Therefore the demigods purposefully say, satyātmakaṁ tvāṁ śaraṇaṁ prapannāḥ: "Real protection can be obtained from You, O Lord, and therefore we surrender unto You."
(Srimad Bhagavatam-----10:2:26-----purport).

"One who cannot deliver his dependent from the path of birth and death should never become a spiritual master, a relative, a father or mother, or a worshipable demigod, nor should such a person become a husband." Everyone naturally gets a father and mother at the time of birth, but the real father and mother are those who can release their offspring from the clutches of imminent death. This is possible only for parents advanced in Kṛṣṇa consciousness. Therefore any parents who cannot enlighten their offspring in Kṛṣṇa consciousness cannot be accepted as a real father and mother. The following verse from the Bhakti-rasāmṛta-sindhu (1.2.200) confirms the uselessness of serving ordinary parents:
laukikī vaidikī vāpi yā kriyā kriyate mune
hari-sevānukūlaiva sa kāryā bhaktim icchatā
"One should perform only those activities—either worldly or prescribed by Vedic rules and regulations—which are favorable for the cultivation of Kṛṣṇa consciousness."
(Sri Caitanya Caritamrta-----3:13:113-----purport).