Businesses have the right to refuse service. If you try to get a table at a nice restaurant with no shirt, they'll ask you to leave.
It's the theater's right to refuse kids to see rated R movies. They can card people for PG movies if they wanted, it's their theater. Personally, I wouldn't let a 13 year old see a movie like that, but that's just me. If she doesn't like that policy, she can take her business to another theater.