You should both be paranoid. Get tested, both of you.
Regardless of who she goes to to get on birth control, her medical treatment is confidential and it's against the law for any medical professional to notify her parents, even if she were a minor. To be on the safe side, maybe she could go to planned parenthood or some other women's center where none of the staff would know any members of her family. In other words, avoiding the family doc might not be a bad idea.
Worst case, it's possible that if she pays for the visit through her health insurance, a statement about the insurance claim (provider, date, charge and co-pay, etc) may be sent to her house, addressed to the subscriber (likely one of her parents). So they may find out that she saw a doctor. It won't say she got on the pill. And 18 year old girls should be seen by a gynecologist on a yearly basis for a checkup anyway, sexually active or not, so she can make that excuse. No one has to know the details unless she spills it or leaves her pill pack around.