If given permission from one parent, can another parent legally move states?

You must either have written permission from the other parent or a court order allowing you to move out of state.

Yes, if both parents are consenting, then a parent can relocate. It should be in writing, signed and notarized and provide all pertinent detail, such as what timesharing or visitation the non-relocating parent will have once the other parent moves. It would be best to have an attorney draft the document so that it complies with Florida Statutes and to then have the agreement filed with the court and ratified by Court Order.