Can a parent have their 10 year old daughter sterilized, legally?

No. It has to be HER choice, and no doctor will do such a thing to a child. Very few will do it for an adult woman who isn't in her late twenties, married, and already has children. A good doctor would call the authorities on any mother seeking to remove her child's reproductive choice from her. You also don't get to decide what your child is "meant to do". I hope she grows up to be an atheist. With TONS of babies.