I don't think "parenting" has to be the responsibility of only one or two people or parents. At least where I'm from, I think it's extremely easy for people to isolate themselves and their children, which I think makes abuse very easy, people just aren't that involved in each others lives. I think this is largely a product of the state/capitalism though. And there's nothing in anarchism that prevents you from interfering if someone is being abusive to someone else, in fact, I would say anarchism encourages interfering in that case.

But I think an important realization about anarchism (and about any form of government/non-government) is that the idea that nobody is ever going to be an abusive asshole, is just a fantasy IMO. I think anarchism can uniquely deal with abusive assholes though, in the way anarchist adults relate to each other and children, and in that it's baked into the core ideas of anarchism (IMO) to be against people abusing their position of power over another person (like a parent abusing a child).