What about in the case of torture? You hear in the news a lot about particularly awful people who torture and kill their children slowly (if it be starvation, actual dungeon-like torture, etc.), is the punishment any different?

From my knowledge, physical punishment on children is only allowed when they disobey Allah Azza wa Jal (like not praying, eating haram, etc.).
But is there any legal punishment on a parent who transgresses these bounds and is outright abusive (emotionally or physically, mild or brutal)?

by

1- Yes those who torture their kids and then kill them, they are punished severely. They are not killed but they are punished severely. Yes if such a parent is deemed as Mufsid fil Ardh (see 5:33) sue to repeated offenses then they can be killed.

2- Parents can only discipline their kids in limited cases and they cannot leave any mark on their bodies. It is very light.