Can a 16-year-old and his pregnant girlfriend move into an apartment owned by the girl's father if the parents of both minor's give their consent?
In this situation, the answer would be yes, because the parents have given permission, provided that both of them have reached the age of consent in their state or jurisdiction or there are exceptions in the law that wouldn't make it illegal.
In most states, any 18 year old is considered an adult and is capable of deciding when and where they reside. They have the ability to contract for housing. SOME states allow minors to become emancipated under specific circumstances, but this is not just a matter of filling out some paperwork.
A minor is the responsibility of their parents until such time as they reach the age of majority or are legally emancipated. The parents are required to provide for the support of their minor children. If the parents give permission, the minor can live in another location. Such permission does not relieve the parents of the responsibility to provide support.