But even an imbalanced person can know killing is wrong.
If you are sick, but you have loving character . . . "Love does no harm to a neighbor," we have in Romans 13:10. So, I would say a person only physically sick but with kind character is not going to hurt someone while only physically sick.
But now we see the lines muddled between physical sickness and spiritual evil of a selfish and worldly personality.
So, if the guy is spiritually sick because of sin's evil spirit (Ephesians 2:2), yes he is liable.
And as much as his parents did not bring him up to love . . . were not good examples, I mean . . . they shared liability somehow, I would say.
But you can't say they were guilty of actively causing him to kill.
Any person born in sin can be guilty of murder. That is on the person who is guilty. But parents are guilty of not being a good example and not teaching the person, in any case.
God told one prophet that people's blood would be on him if he did not tell certain people to repent. So, yes I see how God could hold parents responsible, as much as they did not be good examples and teach and warn their child.
And in the early scripture, if you are told your ox has been aggressive, but you don't restrain it and it kills someone . . . that is death penalty for you. So, possibly those parents got off with a very lenient penalty, if they were told he is dangerous and did not take effective measures to control him.
But this world lets people get away with evil stuff. They watch out for their own, in different ways. So, they might hold the parents liable, but give them what really is just a slap on the wrist.