As a teen, you have the right to consent for your own STI testing and treatment without your parents present or signing off on it. This means you can have treatment without your parents' consent. However, the age at which a minor can consent for STI testing varies among states. In Maryland, for example, any one under age 18 can consent for themselves to get STI testing and treatment, and your doctor is not required to tell your parents about the treatment you receive.
If you are concerned about your parents finding out, the most important thing is to let your doctor know about this concern. As a teen you are legally allowed to talk to your doctor without your parent in the room, and this is when you should voice your concerns.
If you test positive for an STI, there are a couple ways you can get treatment without your parents knowing. You can order medicine online and have it shipped to you (or a friend's house). Also, it doesn't take too long to get treated (for example, Chlamydia is just 2 pills a day for a week), so you can take the medicine in secret if you need to. You can also prevent your insurance company from sending a statement to your parents - find out how here.
For places that are sliding scale and don't require insurance, see this Violet page.