There are a lot of variables to consider when it comes to whether or not birth control will show up on your parents' insurance bill. The first thing to consider is what type of insurance coverage your parents have. If they have private insurance, then it's likely that their insurer will cover the cost of birth control. However, if your parents have public health insurance (such as Medicaid), then there's a good chance that birth control won't be covered. In this case, you may need to pay for birth control out-of-pocket or look into getting financial assistance from a government program like the Family Planning Benefit Program.
Another factor to consider is what type of birth control you're using. Some methods, like the pill, are typically covered by insurance while others, like emergency contraception, may not be. Additionally, some insurers may only cover certain brands or types of birth control pills. So it's important to check with your parents' insurer to see what is and isn't covered before making any decisions about which method to use.
Finally, keep in mind that even if your parents' insurance does cover the cost of birth control, there's a possibility that the charges could show up on their bill (depending on how their insurer processes claims). If you're concerned about this, you can always ask your doctor or pharmacist for a prescription that says do not bill patient. This way, you can be sure that the charges won't end up on your parents' insurance statement.