Could a sixteen-year-old girl move in with her 17-year-old boyfriend and his parents if it is all right with his parents?

No. That's why a 15 year old girl has her own family.

not without the consent of the girls parents. the 15 year old can not legally leave home without her parents saying it is ok until she is 18.

3 ways to do that. 1. permission of parents. This of course is revocable at their wish. 2. Emancipation. The parents have to sign this for it to be legal or a child must go through the courts to "Sue" for emancipation. IE... the parents have to be deemed "unfit". 3. Marriage. Which of course can be granted through the parents. This of course is supposedly why a teenage girl would want to live with her boyfriend.