Whose responsibility is a kid's behavior if not his/her parents'?

Being a parent is a big responsibility, and (barring rare exceptions) you don't become one unintentionally.

Unfortunately, I see plenty of people out in the wild who act like their kids' behavior is none of their business. There was a time when parents unable to control their children would be asked to leave public establishments, but now, everyone is supposed to stand by impotently, and anyone who dares speak up is accosted by the bad parent(s).