Who controls the contact picture on iOS in address book?
Yes, it's possible. If you go to the iOS settings app and sign in with either Facebook or Twitter, there is a button to retrieve contact pictures from those services for each contact on your device. 

So, your child has the option to change that photo on a case by case basis, but what they said could be entirely true. 

At is point, it gets less clear with syncing your address book with Facebook (or other). Presumably it could use each person's Facebook picture, set by them, as their photo in your address book. 

I've never seen a picture I assign get overwritten, but it's clearly possible that might happen.