Douglas Wilson provides his view of parental responsibility in educating children rather than relying on a church or school:

How does the Bible’s teaching on humanity’s sinful condition apply when we come to the education of our children? Children are not free from the inheritance of sin from our father Adam.  In the Scripture, we do not find a contrast between innocent children on the one hand and sinful adults on the other.  Rather, the contrast is between immature sin in the child and mature sin in the adult….

The effects of the fall are clearly visible in young children…What does education do when human sinfulness is overlooked?  What do you get when you educate sinners?  The answer is simple enough – clever sinners.  Knowledge by itself, does not make people better; it may make them worse (Romans 3:20, 5:20).

Some of the most tragic stories in the Bible concern parental failure.  1 Kings 1:5-6 states that Adonijah had unduly exalted himself in aspiring to the throne.  It says also that” …his father had never interfered with him by asking, ‘Why do you behave as you do?'”  David also failed as a father with Absalom, and his son lost his life.  It was the same with Eli and his sons.  They were evil, and Eli did not restrain them (1 Samuel 3:13).  They too lost their lives.  Samuel, who had been used by God to rebuke Eli for being a poor father, failed in the same area himself.  His sons took bribes and perverted justice (1 Samuel 8:3)

Such examples show us that just being around godly people is insufficient.  There is no such thing as an automatic transfer of wisdom.  The children of Christians have great privileges (1 Corinthians 7:14), but it is crucial that we understand the nature of those privileges.  To belive that children can be spiritually changed by their environment alone is behaviorism, not Christianity.  Parents are held accountable by God for how their children turn out.  For example, elders of Christian churches are required to have well-disciplined and believing children (1 Timothy 3:4-5, Titus 1:6).  But that responsibility cannot be exercised well if parents persist in acting as though education and environment alone are capable of transforming the sinful heart of a child.

Children, like the rest of us, are by nature objects of wrath (Ephesians 2:3).  Those who discipline and teach them need to understand this fact, and they must recognize that how a child is brought up has eternal consequences (Proverbs 23:14).  In addition, a child’s upbringing has temporal consequences (Ephesians 6:1-4).  That is, how a child is brought up affects his life here on earth.  It is up to the parents to require the child’s obedience, which in turn is blessed by God.

(Recovering the Lost Tools of Learning, pp. 71-72)