This is for a little dude like Mushu from Mulan
pathfinder
As a pf GM, i would allow it, but not a fully fledged elder dragon, of course. Well, maybe a very old elder dragon with hip problems, complaining about the youth,that would be fun.
But something small, young, sure. You just need to balance it and if the player gets top cocky, there are always dragon hunters around.
Maybe have a look at the Battlezoo: Dragons 3rd Party book, they did a good job in balancing Dragons as a playable ancestry.
Thankfully—if I'm reading the rules correctly—the stats of a familiar are pretty locked in and not dependent on the species of their pet—apart from the specific abilities they choose. So there's probably no concern about a dragon being OP compared to something more traditional like a bat or cat. Unless I'm mistaking something.