No, the child will be in trouble. The parent wouldn't go to jail but would have to pay for medical bills on the person the child hurt. If taken to court judge might add extra cost of pain an suffering .
It depends on the situation. Danimarie is on the right path. If it was a severe injury and the "child" causing the injury should have been supervised they can be arrested for child neglect. If we are talking teenager then the child is the one in trouble. The laws vary state to state.
A parent is responsible for a child's civil wrongs and would have to pay damages (medical bills etc.). However a parent cannot be charged criminally for the acts of a child so they cannot be convicted and jailed.