Paarthurnax gave Zuko a ride into the Bastion when she first arrived, but it seems sort of impolite to ask a sentient being if you can ride them unless they offer first.

"That's right. But don't worry, you'll have plenty of time to ride around on us first."

Well... She really doesn't know how fast this dragon is going to grow up, but she's just so tiny right now. There's got to be a lot of shoulder-perching in her future.