I don't know who Mellissa Harris Perry is, and I don't know the context of what she said, but I agree with this statement to some extent.
What she seems to be saying is that it's not just parents who are responsible for a child, it's the whole of society. (Am I right?)
This is a nice sentiment. If you saw a lost child you would help them, right? If a parent is abusing their child and somebody else, a neighbour perhaps, becomes aware of this, then they should have a responsibility to the child to help. Society should have every child's interests at heart, and it should legislate accordingly.
However, ideally it's the child's parents who should be responsible for them primarily. I guess society should be there to step in if something goes wrong.
The part of the statement I'm not so sure about is children "belonging" to anyone. Nobody "belongs to" anybody else. Our children are not our possessions. They are not things - they are people with their own minds, their own rights and ultimately must make their own decisions. Everybody belongs only to themselves.
I once had a landlady who used to say "your children don't belong to you, you just borrow them for a time". We look after our kids for a while. We are responsible for them when they are young, support them and ideally surround them with love. Then they grow up and make their own lives.