I am sure 81 sounds great............though reality is once she is 18 it will be hard to stop her. As far as who is right, does it really matter? The fact is you are the parents and must pick your battles and decide what is best for your child.
In the end try looking at it as a accessory. Piercings are not forever and to kids its like any other accessory such as a purse or a pair of sneakers. Its something that makes your child feel better about the way they look as well as boost self esteem. It often brings extra attention from their friends and is something many children want.
Based on the above you have to ask yourself why do you not want the piercing? Is it because it is different or because of what others will think of you as a parent? If so is that the message you really want to give your teen? You want your teen to be concerned about her own opinion and not those of her friends, neighbors or loved ones. By saying No just because other parents or family would not agree you are doing the one thing you do not want your child to do.
I am not saying she is right and you are wrong.......I am saying to sit down and talk with your wife about why you do not want her to get the piercing. Make sure your reasons are good ones and then talk with your daughter. If after thinking it though you realize this really isn't a big deal then make up a game plan. Let her know your limits such as where she can wear it and how many piercings she can have. Make ground rules about what type of ring she can wear (Nothing long that can get ripped out).
If you have decided it just is not acceptable then sit down with her and ask her why she wants it. Ask her to be honest and consider what she has to say. If you refuse to give her the chance to explain why she is going to feel like you are being controlling. By giving her a chance to explain her reasons you can explain why you do not want her to wear one.
When you tell her no list your reasons. Here are a few more common ones you can add as well:
Infection is common and can leave a scar
Most schools forbid them in class (Which means she would have to take it in and out......though this will not work if your school has no rules on this)
Employers often refuse to hire teens with piercings or require them to be removed during work.
Should she get into a alteration with another child there is a chance of this being ripped out and could leave a scar
You can use the above excuses as well as your own reasons to help explain why this is not a good idea. Be prepared for some serious begging. Some parents also use these are rewards.......such as a contract. In other words the child must keep a certain GPA or certain behavior in order to get one and keep one.
In the end this is a part of childhood. I have 4 boys and my oldest asked for a ear piercing when he was 13......my first thought was NO WAY! After considering my reasons I realized I was more worried about what the family would think or my friends. So I went with it. Set some ground rules and allowed the ear ring. Less then 6 months later the ring came out and never went back in. Since he was in sports he had to take it out for sports, he had to remove it for school and eventually he started forgetting to put it back in and eventually it faded away like any other fad. Look at it this way. It is not permanent, and at least she isn't asking for a tattoo