Who's fault is it anyway?

The question of: to what degree a parent- or more applicable to me, a father - is responsible for the actions and the spiritual life of his children has come to the forefront of many of my thoughts. In speaking on this matter I believe it is important to know that Christ alone saves, not mom and dad. And that if the Authority to save were to be ours, that would diminish what was done on the cross and the power of the Holy Spirit. That being stated, there is a weighty aspect of the conversation that can be likened to the yield of a crop. Does the harvest not speak to the planting, and the planters? To what extent can a parent be held accountable for the actions of their children? Are they responsible for everything the child does up until the age of 18, up until they move out, up until marriage, and responsible for nothing after?

This is two-fold, children are responsible for obeying their parents, parents are likewise responsible for teaching said obedience -

"Children, obey your parents in the Lord, for this is right. “Honor your father and mother,” which is the first commandment with promise: “that it may be well with you and you may live long on the earth.” And you, fathers, do not provoke your children to wrath, but bring them up in the training and admonition of the Lord." - Ephesians 6:1-4

 Knowing parents are responsible for the training of children, the next question that naturally presents itself goes something like, "But is it my fault if they stray?" 1 Samuel provides a helpful example of this concerning Eli and his sons.

"Then the Lord said to Samuel: “Behold, I will do something in Israel at which both ears of everyone who hears it will tingle. In that day I will perform against Eli all that I have spoken concerning his house, from beginning to end. For I have told him that I will judge his house forever for the iniquity which he knows, because his sons made themselves vile, and he did not restrain them. And therefore I have sworn to the house of Eli that the iniquity of Eli’s house shall not be atoned for by sacrifice or offering forever.” - 1 Samuel 3:11-14

It is evident here that God judges both children and parent. Eli's responsibility after having raised and taught his sons to fear the Lord was to restrain them from living in sin.

"We may learn from Eli how cruel paternal laxity is, and how fatal mischief may be done, by neglect of the plain duty of restraining children. -- He who tolerates evil which it is his province to suppress, is an accomplice, and the blood of the doers is red on his hands." - Alexander MacLaren

Fault is not a prerequisite for being held responsible. Laxity can be likened to the sin of passivity, and we know from the very beginning what comes from Passive men.