I think it’s kind of like Simone’s link on favoritism yesterday. Except, instead of being the child that the parents like the most, it’s the child the parents dislike the most. In a weird way, you could even say that child is the favorite, because the parents deem that child worthy of so much attention.
It’s a scapegoating technique, and it allows the parents to say that if they can just fix this one child, then that will be a panacea. I think that often they don’t explicitly blame the child, but rather something the child has – like, the child just has a bad attitude, or has ODD, or has ADD, or has bipolar, or has an addiction (because mental illness is so “in” right now, as opposed to, say, witchcraft or having TB), and then all they have to do is cure the bad part of the child, and everything’s better. Course, then they have trouble seeing the child as anything other than the bad part, so the child identifies as “bad” not “has a bad part”.