If you file something under their insurance, they will find out about it. They get an explanation of benefits form in the mail for everything the insurance company pays, so there is no hiding it. You'll need to pay out of pocket at your GYN's office or go to Planned Parenthood (clean, reputable, professional, cheap or free depending on your income level) to get birth control.
Regarding the choice of an IUD, though... IUDs are not recommended for women who have never been pregnant. Since you don't want your parents to know you're seeking birth control, I'm guessing you don't have kids. You may have a tough time finding a doctor who is willing to give you an IUD. There are numerous other kinds of birth control out there, though, and some are as low maintenance as an IUD, like Implanon.
Kudos to you for wanting to protect yourself and making the steps to do it. Good job. After you're on BC, be sure to continue using condoms to protect yourself from STDs.
EDIT: Okay, I have to respond to Miss "Nulliparous Status." (Duly impressed, by the way.) The manufacturers of both Paraguard and Mirena do not recommend their products' use for women who are... wait for it... nulliparous. It's not because of difficulty or pain during insertion. It's because they don't want the liability if a woman has fertility problems later in life. While studies have not shown any deleterious effect on future fertility, they want to know that a woman is capable of getting pregnant, simply because they don't want the woman blaming the IUD if she should have problems in the future.
I'm not saying that IUDs are bad for nulliparous women, just that she may have trouble finding a doc to insert it off-label, because most docs don't want the liability, either.