I completely agree with many of the above posts.
1. If you are 18 and mature enough to get something marked on your body forever, you are mature enough to discuss your plans with your parents.
2. Think long and hard about the location/content of this tattoo. (As others have mentioned, employers may not be so keen on visible tattoos, limiting your options later. Also think about what it means to you.. some people don't care if they get a tattoo just because it looks cool - not saying that is the case - but for me, none of my tattoos are just arbitrary designs.
3. If ever questioning your decision to get a tattoo, please hold off. If in several months, your design and location have not changed, and you are POSITIVE that it is what you want, then proceed. Your wrist will be there as long as you are, there is no rush to get it done now.
4. If your parents say no, you should respect their wishes. You are living in their house, and they are still providing for you. If tattoos are something they absolutely forbid, you are better off waiting until you can support yourself, unless you think that a tattoo is worth upsetting your parents, and any possible consequences that it could mean for you (there was a thread not 2 days ago on the CB , some parents posted they would no longer support their child financially if they decided to get a tattoo against their parents' wishes)..
Again this is something you need to ABSOLUTELY discuss with your parents, and give a lot of thought to before you go through with it .
PS The wrist is not an ideal place to heal a tattoo esp during summer - no sun/sunblock/tanning beds/swimming for 2 weeks.