Generally, the doctor will look at your medical history, and find out like your history about your periods, and talk about the main reason for you wanting to be on the pill. Some pills are better for those girls who want to treat their acne, others control the actual period etc. I think when I went, my doc just said ok we'll start you on Levlen and ive been on that since.
in regards to seeing the doctor, you could just for hte moment borrow your parents medicare card and tell them that you need to see a doctor, just make up some excuse. And while you are out, go to your nearest Medicare office, and ask them for your own Medicare card. It doesnt cost you anything. Your parents will get a new card I think as well, without your name on the card, or maybe you can elect for your name to stay on their card.
I had the same fear as you, going and getting the pill behind mums back. When she found out that i was hiding it under my pillow, she blew the roof off nearly lol. Which is quite strange since she's on a similar thing i think. But eventually she finally listened to my reasons for going on it. She prob mainly thought i was having sex, which is not at all true, I went on it to control my periods, even though that didnt work until late last year after I had that operation.