According to the Bible, who is responsible for the rearing of children?

I don't see an exact reference to the scripture in this chapter;

For Hester Prynne’s sake and for the sake of the young child, let us leave them as God has seen fit to place them!”

Children belong where God has placed them.

Ephesians 6:4 / New International Version (NIV)

Fathers,[a] do not exasperate your children; instead, bring them up in the training and instruction of the Lord.

[a] Or Parents

The Bible states in Proverbs(in chapters around 13-17)

"A Wise son accepts his father's discipline

A Wise son makes a father glad"

(Yes, its not the exact one, its just parts and bits put together)

It also states that a father is supposed to be the head of the family, making sure everyone in the family is "in check".

So your answer would most likely be father.

SIDE NOTE:

Believe it or not, who ever is responsible in raising the child is a controversy when looking at the bible's standpoint because if there is only mother to take care and raise her children, is that her responsibility now? Of course, my church and several other friends had a controversy about it, not sure if over 25% of the world has this kind of controversy.