What everyone else said
AND
Go to the library. Get out all of the books that you can about eating vegan and vegetarian. Including cookbooks. Take them home, and read, read, read. Take notes. Write a report. Show your parents your report, and your source material.
I'm guessing that your parents will be more impressed with information from published books than from the "anyone could have written that" internet.
Also, if you have a teacher at school who teaches nutrition or a cooking class or health, talk to them, see what they say.
Oh, and if you can (and I know it's incredibly difficult) perhaps you can ease into going Vegan? Make a deal with them, if you eat a burger on Thursday, you can drink all of the soy milk that you want on Friday. I know, it's crappy. So many people are not just scared of what will happen to people who don't eat meat, they're also scared of what will happen to people who eat the dreaded soy. If your parents can see that you're healthy on the alternatives, they might let you phase out the animals.
Also, make sure that you aren't giving your parents any reason to believe that you are starving yourself. Pig out on vegan food!
It can be difficult; my grandparents were convinced that my sister was anorexic because at one meal she didn't want seconds of dessert...