Obviously I don't know what would work with your parents, but here's what
my friend's son did to convince them to let him have a cornsnake as a pet:
He wrote up a actual "proposal" paper which laid out all the information.
-- Why he wanted a cornsnake
-- Lots of cornsnake information (temperament, feeding, housing, handling, etc)
-- How (step by step) he would care for the snake, including paying for all the
costs involved (this would be especially important for a dragon...they cost a lot
more than a cornsnake to care for properly...feeders can get expensive, plus all the
other initial and on-going expenses for enclosures, lights, heating, vet costs, etc)
Basically he showed his parents that he had done enough required research to be able
to correctly care for the snake, that he had read up on and understood both the responsibilities
and the costs involved, that he was mature enough to accept the multiple-year responsibility
of caring for an animal, and that he was still wanting to have one after doing all that research.
Next, he got them to take him to a local reptile expo. This let them see other reptile enthusiasts
with their animals and allowed them to interact with some of the reptiles themselves.
[this can be a big one -- having them interact with a beardie can often make a big difference
in their opinions about their personalities and temperments]
After all that, his parents decided to make him wait for at least 3 months (and to show that he
had saved up enough money for the initial costs and could continue to sustain enough money
for the on-going costs) and if he still wanted one after that time then he could get one (which he did).