Now, you can't really tell your parents that your mature enough, you have to show them that your mature enough and let them realize that you can handle it.
First you should do some research on the different piercers in your area. Find the best one. See how clean they are and everything.
Next, research how to clean an eyebrow piercing. You need to know that it could get infected, but you should know how to take care of it if it does. Clean it often, 3 to 5 times a day, don't touch it without cleaning your hands first, don't turn it without cleaning all the goo off the ring first, etc... You should realize that puss will come out of it for a while, but that's normal, and don't freak out. You should know what to do when it gets too dry, by putting A+D ointment on it or even bag balm or something like that. Search online and find a good website to help you take care of it
Next, you have to realize that your parents may have some bad images of people who get piercings. Let them know that not all people who get pierced are "bad". I've seen many people who tasteful visible piercings who have very good jobs, even when it comes to working with the public. And, as you get older, if you choose to take it out, you can without a visible scar. I've got two facial piercings when I was younger, and took them both out, but no one would know it unless I told them because there are no scars. That is one of the missconceptions about piercings. If you go to a GOOD piercer, who does things right, then you will have a very clean piercing that can heal good if you ever choose to take it out.
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