Hmm. I would say "Mom, Dad, if you wont let me get a pet snake, can I get a Betta? I will take care of him myself and buy everything he needs. You will have to do nothing but take me to the pet store.."
But, caring for bettas is much more different than you would think. Check out our "General Betta Care" sticky for info on what you need/what you need to do.
Make up a list from the sticky of what stuff you need. Then look at it, say to yourself "Self, am I capable of buying all this stuff right away?".
Then make a list of what you have to do to care for the Betta. Then say to yourself "Self, do I have the time and effort to feed my Betta every morning, to test the water and to carry water for a water change? Do I have the patience to wait a month through the cycling process (which is a great time to learn some stuff about the nitrogen cycle)?" When I was a kid my mom had no idea how to take proper care of fish and I learned bad habits from it so it was doubly as hard to actually try to do things right when I journeyed back into fish keeping.
Ask them for a Rhinocerous. They'll say no. Ask them for a Komodo Dragon. They'll say no. Ask them for a Grizzly Bear. They'll say no. You can repeat this for a while. You know your parents best. Just keep throwing ridiculous pets at them until they get frustrated - kinkajoo, wallaby, Crocodile, Spotted Owl, wildebeest. Then say, "Geez, guys. Well, if you won't let me have any really cool pets, can I have a betta fish?"
Just make sure they won't say yes before you're ready - so don't ask them for a timberwolf unless you will settle for a husky or some other sort of wolfish dog. Also, mix up the type of animal. If you keep pestering them for reptiles, you might come home to a tank with anoles some afternoon.
what might work (but i do not recomend it) just go to the lfs yourself and get the fish, tank, etc, without them knowing. you'll probly be grounded to months when you return home with a new pet. but they'll get over it. (if they dont make you take it back) again i dont reccomend this unless your parents are the type that will work with this.
My brother was 14 and he wanted to keep fish. I let him take care of my tanks while I was on vacation (back when I lived at home), and he refused to do water changes, claiming he didn't know how, even though he did. My parents concluded at that point that he wasn't allowed to have fish.
HOWEVER, doing the opposite is a good way of proving you should have one. Do lots of research on bettas and show that you know how to care for them. Also, tell your parents how beneficial fish can be. How it is good for you to learn responsibility and caring for a creature dependent on you and how calming and relaxing it is to have any hobby, especially fish keeping. The most important thing is for your parents to realize that they won't have to do any of the work. Kids are a lot of work without taking care of their pets too!
u could..........*whispers* steal one, break into a couple of houses, although ur 13 so u might want something to pull the tank on, espeially if u are going for a big one.................oh and drain the water out before u take it otherwise its impossible to carry.........watch the show "It takes a Thief" practice at ur house, then do work son, nah dont do that, im just joking around, just rake some yards and buy one