i'm trying to tell you not to get so hung up on names. i mean you say you don't care where you go to medical school but your parents want you to go to a good school. the only thing with that is not only you but your parents as well probably don't even known of most of the medical schools by reputation except the IVY's and your state schools. In the end there are a lot of considerations when choosing the medical school you're going to eventually end up at (assuming you get accepted). Some schools are more research-oriented, some are more primary care, and some have excellent specialties. There's also location, tuition, culture, quality of facilities, and etc.. The number one ranked school may not be your first school. You just have to find one that fits. As you go on interviews, you'll discover that a lot of places are actually really nice and I personally could see myself happy at a lot of them. The only times I were unhappy was just b/c the facilities were a little old or the school was located in a poor neighborhood. These were the things I thought about and not rank. I'm sure if your parents come with you, they're going to feel the same way. Right now my parents just want me to go to a UC for in-state tuition and to stay close to home.
Are you good enough for top 20 schools? I don't know enough about your stats especially your MCAT and your life experiences. However, I will say they have to be very substantial and nothing about what you've done so far or probably can do is going to change that. That said, there's no harm to add a few reaches into your list of schools b/c everyone does that and who knows you might get lucky or someone will see your true potential.