You should go to a place that 1) is a "real" school and 2) is cheap. You'll be kicking yourself if you go into medical school with $100k+ of debt. I don't care if it's Harvard. If you go to a reasonably large institution, it won't hold you back.
Ultimately it's your decision. Unless your parents are paying everything, I would respectfully tell them that you appreciate their opinion, but it's your decision.
I'm guessing actual medical students, not premeds, will be more inclined to give this advice. For everyone who says that you should get into the best [ranked] university possible because adcoms care, this may be somewhat relevant when we're talking about a top ranking school, but even then it will still fall bellow your MCAT score and GPA in importance, and very likely bellow your extra curricular activities. As far as real advantages of higher ranking universities, as far as I can tell, are limited to 1. Research opportunities and access to brilliant faculty, 2. being surrounded by smarter and more motivated peers (also more competitive which can be a bad thing for GPA), 3. as mentioned earlier, you would get a life experience.
I did pretty much what you want to do, I stayed at home and went to a nearby school even though I got accepted into a better, more well known, state school that was a few hours away. I did it to save money (parents covered my undergrad and home expenses, they wouldn't have covered dorms and living away expenses), and stay with my close friends, who all also stayed in town. As others have said, you will be missing out on a life experience - living away from home, dorming, and for a while I did have doubts, and regret over my decision, but I see a lot of reasons now why staying was a good thing. I didn't have trouble doing well on the MCAT, or getting into my top choice (not a top ranking medical school).
I had never intended to be an MD/PHD neurosurgeon who went to Harvard medical school and did my residency at John's Hopkins. I don't have a competitive personality, and even though I might want to do a competitive residency (ophthalmology), I certainly don't need to get into a top residency program to become one. The ophthalmologist residents who are at my school's residency program largely went to my school, and they made a point to say that in deciding who they take, they can care less about what school you went to, they only care to see how interested you are, and how competent a doctor they think you will be. If however you do want to do neurosurgery, or derm, or go into academia, it would be very beneficial to go to a top school, and get into a top medical program, and then go into a top residency program, all while getting published under brilliant (and well known) researchers. Frankly, the attitude some people have here screams gunner, and if that's you, then you shouldn't hold yourself back, but if you're laid back like me, and don't need to be a super star, then don't feel pressured to go to the best schools and be the best student.