This is a hard subject to say yes or no on but I’m going to go with a No.
Why children are criminals depends on many different things, how they are raised; peer pressure; mental illnesses; etc.
Let’s say the child is being raised as a good citizen and a good moral person as I’m sure most are, but committed some crime due to peer pressure. Then in my opinion (this may seem harsh) the parent should not be held responsible at all. To make the parent responsible for the children’s acts of crime would be in a way teaching the children that their mommy and daddy will bail them out or will take the blame for their bad decisions. If my son got in trouble I would tell the cops to take him to jail if need be, or if it doesn’t require jail I would see to it that he does some sort of community service or did some jobs for neighbors to pay off the fine. I believe that in doing this I would teach him he is responsible for his own actions and that there will be consequences if he chooses to do something he knows is wrong. I would in no way, shape, or form bail my kid out of jail or try to talk to the police to get him out of trouble. However as far as fines and tickets go I would pay them up front BUT, like I said I will make my child do community service or several odd jobs to pay me back. In this case parents should NOT be held responsible.
If the parents are letting their children run around unsupervised and letting them act like little hellions, then I say to the parents GROW UP and actually be a parent, children need guidelines, rules, and boundaries. They will have enough friends they need an adult. On this point the parents should be held responsible.
If your child has some kind of illness and you have tried everything possible and they are still acting out, it may be time to look at institutions or several of the homes that have a 24 hour watch. It is hard I’m sure but it may be best for the child and everyone around him/her. On this note it is hard to say, however if my son acted this way because of an illness, I would definitely help him out and take the blame, he couldn’t help it.