Ok, there is no such thing as stepparent rights. However, it is possible for dad to designate his wife to represent him via POA in dealing with medical matters with his children.
Mom does not have to add stepmom to the HIPAA form at the pediatrician's office, however dad might be able to add her later.
My personal opinion, which has nothing to do with legalities, is that any parent who tries to insist on their new spouse being added to things like that is being a disrespectful A hole. However, again, my personal opinion has nothing to do with legalities.

