"they just want to know where I am for their peace of mind"
So why can't you accept this? when you are a parent, you will know what it is like to worry about your kid.Granted you DO have a phone and if they wanted to know where you were, they could just call you.
"They'll do this sometimes just to make sure I'm where I said I'd be, and will get on my case if I'm somewhere else"
Then perhaps obeience and honesty will give you more freedom.
"I was diagnosed with a medical problem last year, and they're using that as their excuse for being overly-paranoid"
Or, they actually care about you and you are trying to down play your condition because you are fustrated.I see that you are a top contributer to cancer and as you mentioned, you were diagnosed with a medical condition...assuming cancer?
Just look at this situation as if the script was flipped.They are already freaked out that you have cancer, their baby girl and the fact that you are roaming about is unsettling to any parent to a teen because the world isn't exactly fantastic.
I think with all the news reports of missing kids, attempted kidnapping, parents are getting more and more worried.This is why I am enrolling each of my children into martial arts so they know how to defend themselves.
I would simply ask your parents for a compromise.If they want to know where you are, you have to promise to answer your phone NO matter how many times they call and it will make them feel a lot safer.
In the meantime, if you don't like it, I suggest buying a phone and paying the bill your own self because "beggars cannot be choosers".