The whole main part of having a child is responsibility. Parents have to keep control of their child's health, education, and happiness. They shouldn't be too strict though. If they need to keep an eye on them, and they need to support them. Besides, children are still minors. They can't just take care of themselves in a world full of criminals, economy, etc. Parents have to take responsibility if the child runs away because, like I said, they're guardians. It's their job!
Kids run away, because they are mad at their parents, or at least most do; thus its the parents responsibility to ask their kid/kids what is wrong. Some parents give lack of attention, so kids run away to create attention. Now if the kid, runs away, and the parents are nice to them etc... It is still the parents responsibly to check on their kids. If the kids get under drug influence, the parent should find out. Although most kids hate their parents being too obsessive,the opposite is bad to.
When a child is capable of running it is clear that the parents are not in control of the child. It is very possible for youth to be uncontrollable, therefore I find it difficult to hold the parents responsible. Although they may have caused the child to be uncontrollable through bad parenting, they should not be held personally resonsible for the child's actions.