Convincing parents can be a job in itself. You've done the correct first steps in doing your research and collecting as much information as you can about bearded dragons. Now, as I tell my students when we're doing persuasive writing, you need to examine your audience and find out what their positions are so you can counter their claims.
If your parents are hesitant to get you a beardie, start by sitting down over dinner or in the evening and talking to them about it. Start by explaining that while yes, you really do want a bearded dragon, that you know that this is a team effort - you don't have the wherewithal to do this on your own. Ask them what their concerns or hesitations are. They will probably come down to these:
1) Money. This is a big one. Reptile setups DO cost money, and plenty of it, to do it right. Things like proper UV bulbs (which you'll want to order online rather than buy in a store - cheaper and better quality) are a recurring expense; things like the tank are a pretty fixed expense until you want to upsize. Offer to do some research here on how much the setup will cost (there's a great article under "Care Sheets and Articles" about what to buy to set yourself up). Then do some price checking on feeders. You won't want to rely on pet store crickets... or probably crickets, period... when there are so many better feeders out there. But if you get a baby beardie, you will be dropping a SIZABLE chunk of change every month on feeder insects.... it can be truly mind-blowing how much these little guys EAT! You cannot let your parents get surprised by that. I'm on a reptile fan group on Facebook, and recently saw a post from a woman who wanted to get rid of her daughter's beardie because it was costing over $100 a month to feed. You need to prepare for that, and be ready to offer to get, say, an adult rescue instead of a baby to mitigate the costs.
2) Bugs. Many parents are not thrilled with the idea that you will need to keep large quantities of bugs in the house, and perhaps be breeding them to save cash. You'll need to find a way around this; beardies need their bugs, especially baby beardies. Some bugs will need to be kept in the fridge to make them last longer. Your parents need to be okay with this - or be willing to offer an alternative.
3) Vet care and upkeep. Dragons should go to the vet at least once a year to be checked for parasites. Locate the closest reptile vet to you, find out what the costs are, and be prepared to share what you will do to help offset those costs. As far as daily maintenance goes... reptiles are generally easier in that regard than cats or dogs; their poos are small and in one cage, and they eat smaller portions of salads than dogs eat kibble. Any responsible kid should be able to handle daily maintenance on their own. But again, you'll need mom and dad to provide you with the supplies... the vinegar and water for cleaning, spray bottles, etc.
Just talk with your folks. Explain that if you have to, you'll wait until you have a job to help offset the costs - but that you'd like to know what it would take from them to make this work. Parents can often remember being young - they will remember wanting something badly, too. They generally aren't going to say no for no good reason... and understanding that reason can give you a way to understand them, and that will make your relationship stronger.
Good luck!
