You have every right to place healthy boundaries in your marriage with his ex-wife. As I was reading your post I was under the impression that they still had kids together and continued to be in contact for their sake and also for coparenting purposes. I would find it completely unacceptable for my husband to keep any type of communication with his ex. Other than children being involved, there is absolutely no reason for any sort of relationship to be continue after a divorce.
Now his parent’s continued relationship with his ex is up to them but I would have a talk with my parents if they continued a relationship with my ex. Only out of respect for my new spouse.