Honestly if your parents don't like paintball it is hard to convince to get a unless you take them paintballing and they like it, but if they really don't like it, you're screwed! Try to show your parents you deserve one by being responsible and helping out around the house. No I am not a parent trying to help other parents, I am a kid and this totally worked for me.
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For convincing parents of anything, you should try explaining the benefits of it to them first. Common arguments. concerns for being able to play paintball are: physical activity, safety and cost. If you can show them this, in an intelligent way, you have a better chance of them saying yes. You could also ask them to go paintball once with you. However some parents simply don't want anything to do with "gun's" then you are simply out of luck.