If you talk to Planned Parenthood about confidentiality, you will be fine. If they call you, they will use code names like it's "sally" or "bill" or something similar so your parents don't know if they answer.
Also, just as a warning, if you use an insurance that your parents use too, chances are it is going to be written on your insurance bill.
Perhaps to avoid an awkward encounter you could tell your mom that your periods are really heavy, irregular, and painful and you would like to get on the pill because it makes your period more regular and less painful. She doesn't have to know the real reason why.
Good for you for wanting to be protected, though. That is wonderful