JOEL - You are right. There is a point in time when our children are on their own and are responsible for their own actions. This story is often used as a lesson on the importance of training our children to be obedient. However, in Eli's case the time for proper training was when his sons were younger.
I believe the main reason Eli was held accountable in this story, was not just because he was their father and should have taught them better when they were younger; it was because he had the power and authority to remove his sons from their office but did not do it, despite the repeated warnings from God. (1 Sam. 2:29-34), He allowed his sons to continue to work at the tabernacle (1 Sam. 4:1-5), the results of which were disastrous for him, his sons, and the nation. This story is probably better used as a lesson in leadership and responsibility rather than proper parenting.

Perhaps a better example of the need for proper child training is described in the D&C:

"But I have commanded you to bring up your children in light and truth.
But verily I say unto you, my servant Frederick G. Williams, you have continued under this condemnation;
You have not taught your children light and truth, according to the commandments; and that wicked one hath power, as yet, over you, and this is the cause of your affliction.
And now a commandment I give unto you—if you will be delivered you shall set in order your own house, for there are many things that are not right in your house." (D&C 93:40-43)

The only reason we as parents might be held partially accountable for wayward children is if we did not do all we should have done when they were young and under our care. (See D&C 68:25)