Honestly I think if a couple isn’t able to have their own place (not talking owning, ok to rent), they should just date. No need to live together and move in with parents.

I’d say household chores you describe are just normal things people do. You don’t do it for her. You live there so washing dishes and cooking for yourself and your boyfriend (sometimes? I’d say it should be all the time). is a bare minimum.

Are you two married? It sounds that you two are not independent yet and are depending on your parents. So you living with your parents and him with his is very appropriate. When you two can manage on your own, then live together