I want to go to Edinburgh, but my parents are reluctant Watch
Don't think this will change your parents mind that much, but you could tell them that as an American, I find the 2 hours or less thing completely mindboggling. The UK is so small and going to Scotland from London is NOTHING compared to traveling in the US... people in the US can be two or three states away at uni at a bigger distance than that!
Also, remember to visit (duh!...do you say that in the UK? Lol... me and my Americanisms). Before I visited I was in LOVE with Edinburgh and thought that's where I'd like to while the rest of my days, but turns out I liked St. Andrews much better (and I only visited St. Andrews by a fluke... I wasn't even really considering it).
I decided to go for one 3 hours away by car. All my choices were about this distance, apart from Durham- but that's just such a nice place, anyway. I didn't want to go closer, as I felt like I needed to gain maturity. The only way I could do that was getting further away.
I would suggest having a talk with your parents. If you go to Edinburgh and find that it really is the right place for you, that you will enjoy, then you will have to convince them of this. Ask them why they only want you to be 1-2 hours away, and do tell them that the lifestyle at uni means that your coming home at weekends would be unlikely. university is partly about growing up and moving away from the home environment- that may be something you need to highlight for them. You would be there not only to learn academically, but to have the whole experience and learn socially- if that makes sense! Living away from parents is one of the biggest challenges you will face.
So my advice is to bite the bullet and ask them why they want you to stay so close.