You can still get a pet dragon - you just have to raise one. Talk to the alchemist and complete his quests, and take note of what he says about feeding the wyrms in the wyrm pen.
I did, I completed the wyrm pen quest.
Oh and, how specifically do you raise a dragon?
Feed a wyrm in the cage until it's full - it will turn into a cocoon and eventually become a baby dragon.
Thanks!