Is it legal for a ny doctor to administer a HPV vaccine without parental consent?

No it is absolutely illegal for any doctor to administer anything without consent to anyone.  It can be considered assault or medical malpractice. You need to speak with an attorney, however, because if your child is of the age of consent (i.e, can make own medical decisions) and you were not in the room, you may not have a case. Think of a 16 year old who decided to have the vaccine and indicated it was fine. Talking to an attorney will help you determine whether you truly have a case or whether it would simply mean you may wish to just file a complaint against the physician with the state medical board.