They cannot simply wash their hands of you if you have not reached the age of majority. They can certainly give you permission to move out. You may have difficulty if you are not an adult in signing contracts.
If a child is legally emancipated by the court, then the parents are no longer responsible for the child. On the other hand, if the parents simply give the child permission to move out...then that's...
You petition the court if you are allowed to by the state's statutes. The Court will take the parents' desires into consideration, but it is not a given.