Hey
I was in a similar situation... my parents were totally against me getting my nose pierced so I had to persuade them to let me.
First off, I printed off a load of information on nose piercings (or whatever piercing it is you want to get) that details the procedure, aftercare, risks involved etc. Alternatively, you could direct them to a site like
www.bmezine.com that gives information on piercings and also member's stories and experiences of their piercings. This shows your parents you are researching your decision and are being sensible about it.
Secondly, as your parents are strongly against piercings as they think it's a sign of rebellion, you need to show your parents that your reason for wanting a piercing is NOT for rebellion - tell them it's for your personal reasons, not because you want to rebel or look different to everyone else. Think of the reasons you really want your piercing and tell them to your parents.... they need to be assured.
Another thing you might like to do is offer to let your parents come with you when getting your piercing (either if they say you can go ahead, or say this whilst persuading!) - that way they can see the standards of the place, talk to the piercer personally about any worries they may have and basically to ease their minds. They might have to come with you anyway, depending on the piercer and if you need parental consent.
The important thing is to try and make your parents feel comfortable with your decision... if you act openly and encourage them to come along, view the piercing studio and things like that it gets them involved and eases their minds.
Hope this helps. If you need any more help PM me or post back here