D&C 68

Recently, there has been publicized an opinion that the burden and responsibility of raising children is more of the community, not necessarily the family or the parents.  Of this thought, President Brigham Young taught -

“If Brother Brigham shall take a wrong track, and be shut out of the Kingdom of heaven, no person will be to blame but Brother Brigham. I am the only being in heaven, earth, or hell, that can be blamed. “This will equally apply to every Latter-day Saint. Salvation is an individual operation. I am the only person that can possibly save myself. When salvation is sent to me, I can reject or receive it. In receiving it, I yield implicit obedience and submission to its great Author throughout my life, and to those whom he shall appoint to instruct me; in rejecting it, I follow the dictates of my own will in preference to the will of my Creator.” (Discourses of Brigham Young, page 390.)

It is clear that salvation comes by obedience to the laws and ordinances of the gospel.  It is an individual thing, not a collective thing.  Much of the commandments and gospel of Jesus Christ includes preaching, teaching, ministering, rescueing, and so forth to all of God's children, especially the family.  But it is still an individual responsibility.  Following this responsibility, the Lord said -

"...inasmuch as parents have children in Zion, or in any of her stakes which are organized, that teach them not to understand the doctrine of repentance, faith in Christ the Son of the living God, and of baptism and the gift of the Holy Ghost by the laying on of the hands, when eight years old, the sin be upon the heads of the parents."  (Doctrine and Covenants | Section 68:25)

In my mind, this does not mean that all the sins of the children will be heaped upon the parent, but that the parents have committed a grave sin in not doing so -

"For this shall be a law unto the inhabitants of Zion, or in any of her stakes which are organized.
27 And their children shall be baptized for the remission of their sins when eight years old, and receive the laying on of the hands."  (Doctrine and Covenants | Section 68:26 - 27)

And not only these first principles and ordinances of the gospel, but all teaching and commandments including -

"...they shall also teach their children to pray, and to walk uprightly before the Lord.
29 And the inhabitants of Zion shall also observe the Sabbath day to keep it holy.
30 And the inhabitants of Zion also shall remember their labors, inasmuch as they are appointed to labor, in all faithfulness; for the idler shall be had in remembrance before the Lord."  (Doctrine and Covenants | Section 68:28 - 30)

The Lord even warned the saints -

"...I, the Lord, am not well pleased with the inhabitants of Zion, for there are idlers among them; and their children are also growing up in wickedness; they also seek not earnestly the riches of eternity, but their eyes are full of greediness.
32 These things ought not to be, and must be done away from among them..."  (Doctrine and Covenants | Section 68:31 - 32)