Well, if you are a. a student, b. under 24, and c. your parents pay half your expenses or more, they can claim you as a dependent as a qualifying child. They can also claim you as a dependent as a qualifying relative after you are 24 even if you aren't a student as long as a. your gross income is less than $3650, and b. they provide more than half your support.
Source:
http://www.irs.gov/pub/irs-pdf/p501.pdf
For the FAFSA, while it is true that students are considered independent of their parents when applying for grad schools, individual med schools may have different requirements. I know that all the schools I applied to (and the vast majority of schools from my understanding) required parental financial information to be eligible for need based aid. So you are probably only independent in the eyes of the service that collects and standardizes your financial information, not the ones actually decided if you need aid or not.