What is the legitimate Islamic solution to avoid any violation of Sharia in the event that the son rebuilds his parents' house at his own expense while both parents are still alive and he has brothers and sisters?

If you agreed with your parents to demolish the old house and rebuild it at your own expense, in exchange for building a private floor for yourself, then you only own this floor, and the rest is your parents' property. If your construction was in the form of a loan to your parents, then the entire house belongs to them, and what you spent is a debt upon them that you are to collect from them. The method of your utilization of the floor must be specified: if it is a rental, the rent and its duration must be determined, and it can be deducted from the debt. If it is a gift, it is not permissible unless they bestow a similar gift upon your other siblings. If it is a sale, the price is determined and deducted from the debt. It is advisable to expedite the resolution of the matter to avoid dispute.