Condoms are only about 80% effective. Accidents happen with condoms – they can break or slip off or something could wrong. So, ideally, you want to be on birth control AND use a condom to be as safe as you possibly can.
If your doctor isn’t someone you trust to keep your confidence then I suggest you google free clinics in your area (such as Planned Parenthood). They will be able to help you with all your questions about safe sex, testing, and birth control needs. And, no, they will not contact your parents. They are required by law to keep your records confidential.