Piercings and tattoos are NOT "self-mutilation". I'll honestly never understand why some people are ignorant enough to believe they are, but to each their own I suppose. I just feel that the judgment is senseless and unnecessary. You don't actually know those people for real, so who are you to automatically judge them and deem them "bad", "dirty", "trouble makers", etc., just because they have piercings and/or tattoos?
Anyway, there really isn't a whole lot you can do at this point. Parents can be VERY stubborn and strict with their rules. Since you're only 14, you definitely still have to listen to what your parents say. Unfortunately, if they say no piercings, it's no piercings. You can try to reason with them like if you get a really good report card, great grade on a project/essay, etc., maybe ask for the piercings(s) you want as a reward. Or maybe wait until your 16th birthday and ask. The biggest thing is don't be a brat about it. Don't yell at them or whine. If they keep saying no, go with it and let it rest for awhile. If they will never budge, you'll just have to wait until you turn 18. At that point, you're free to get as many tattoos and piercings as you want without parental consent.
Also, to the ones saying piercings are permanent - um... no. They are not. All you have to do is take out jewelry and allow the hole(s) to close up/heal. Worst case scenario it MIGHT leave a faint scar. This doesn't always happen though. Tattoos aren't even "100% permanent". They CAN be removed/covered up, but this is often much more expensive and painful than actually getting one. Tattoos are much less forgiving than piercings, so they require much more planning/thought.