Are parents responsible/accountable for the sins of their children? Reading 1 Samuel 2:12 et seq, one could say that Eli is treated the same way as his wicked sons, although he tried to dissuade them from doing their sins.  

Generally speaking, are parents today responsible/accountable for the sins of their (adult, if it makes a difference) children? More than they are in common for their surrounding to tell the gospel?
Disclaimer

I have been studying the book of Revelation, and haven't looked at OT for a long time. Thus, if I mis-remembered the story of Eli, please correct me. Thanks!

Answer

Jeremiah 31:30


  But everyone shall die for his own iniquity: every man that eats the sour grape, his teeth shall be set on edge.


This does not say "die for parents of children's iniquity"

Revelation 20:12


  And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works.


This does not say "judged for parents or children's works"

How then does one explain Deuteronomy 5:9?


  Thou shalt not bow down thyself unto them, nor serve them: for I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me,


My personal understanding is that parents who reject God teach their children to reject God, and thus, the children, in following the parents advice, reject God and thus the sin "punishes for multiple generations."

Furthermore, in:


  Deuteronomy 6:6-9
  6 These commandments that I give you today are to be on your hearts. 7 Impress them on your children. Talk about them when you sit at home and when you walk along the road, when you lie down and when you get up. 8 Tie them as symbols on your hands and bind them on your foreheads. 9 Write them on the doorframes of your houses and on your gates.


Parentes are commanded to raise their children to follow God. From thus, I would personally extrapolate that not raising up one's children to follow God is a sin; to not discipline them when they break the commands of God is sin.

In my personal interpretation, the failure of Eli were:


not raising his children to follow God (though I'm not sure how to handle this since I believe in predestination)
tolerating their evil behaviors (i.e. not removing them from power)