The way I got into stuff like this as a kid was to just nerd it up. If you spend a lot of time on this site and other sites and really become a "fish geek," it will help. Just talk about fish all the time and talk about the nitrogen cycle and the needs a betta has and whatnot. If you can show your parents that you're really interested in fish keeping as a hobby and that you will be serious about it, they'll be more inclined to let you have one. If they just think you want a fish as one of those childish whims, then they assume that you'll stop taking care of it after a few weeks.
As a kid, this worked for me with pets, guitar... all sorts of things.