I'm very happy to see/hear that you're so willing to be responsible like this. You're a smart young lady.
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But about your actual question...You do not need your parents' permission to go in birth control. Since you can drive, contact your local family planning clinic - if you are under 18, you can get anything and everything from there for free, be it birth control, condoms, or even an examination if you choose.
Or if you aren't sure if there is a family planning clinic near you, or you want to find out where the nearest one is, contact your local hospital, ask to be directed to the ob/gyn office, and ask one of the nurses where the nearest family planning clinic is.
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Good luck to you, dear. Be safe, and have fun!
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Scaredbaka