Maybe you can find a tank size you like in your local pet store, or shop online for good deals and put together a "Betta kit." (total cost of items+fish) Seeing as you are finishing another year of school with good grades maybe it could be for a "graduation present." If you chose a 10 gallon tank, and your parents would go for that size tank, it would open up the opportunity to divide it, and have additional Bettas. (The 10 gallon tank size is good for a lot of different uses, and pets, but the 2.5-5 gallon are great for Bettas too.)
Good luck with your parents though. :)
.....Betta Fish are like potato chips, it's hard to have just one! Arm yourself with knowledge. Consider advise. Make informed decisions.
I say you make a proposal to them. Petstores give you a 14-15 day period where you can return the fish, and a 60 day period to return regular items. Ask your parents to let you set up a tank properly before getting your betta, and then getting your betta. Tell them something along the lines of "If you have to do anything at all in regards to caring for my betta because I don't care for him during those 15 days, then we will return my fish." You could mention something like depending on the size of the tank you will be doing anywhere from 3 water changes a week for those two weeks (for a 2.5), to 1 water change a week every week (a fish in cycling 5-10 gallon).
Try explaining to them that your fourteen not a little girl anymore. Also show them that betta's are great starter fish and incredibly hardy provided they have a heater and tell them that you'll do the waterchanges. Its a ten minute job for a 100% change and five minutes for a 50%. Show them that we're all very passionate and if you want PM all the teenagers on the forum and interview us to get our opinions on how a typical day for us and our bettas go.
To be honest bettas are the first fish I've owned since I was seven years old. It's been a learning curve from requiring heaters, to battling finrot and selecting a healthy betta. Plus betta's are incredibly friendly fish and adore people watching. One of my boys was giving my mom the stink eye this morning when she was looking at the phone messages (the answering machine and phone are right next to his tanl.) He scared her since he was suddenly there right infront of her. I hope you can get a betta soon. Try showing them your ready maybe checking out the local petstores and comparing whose healthy, what needs to be changed ect. Just whatever you do don't buy an expensive plakat or HM as your first. Veiltails, Crowntails and females of any tail type are a lot easier. Plakats are jumpers and HM's can be notorious tailbiters.
Lelouch of the Petsmart Rebellion - Orchid VT
Yugi - German Sheperd/Pitbull mix. The real King of the house.