i suggest you get a small dog simply because your parents will probably like that better. my parents refuse to ever get a big dog because they shed and stuff like that.
i have a papillon/shih-tzu and he doesn't shed whatsoever. when a dog sheds you need to brush him on a daily basis and you'll probably end up vacuuming a lot more than you really wanted to. another thing you might want to consider is that if you get a pure bred they can cost up to 700$ and more. but if you get a mixed breed you won't be able to take them with you across the border, or on a plane. and then you also have to consider who's going to look after the dog while you're on vacation if you don't take them. kennels are expensive (my dogs cost up to 500$) and you might not have anyone who wants to take care of them.
i never really proved to my parents i was responsible. i just always wanted a dog. i was watching 101 Dalmations since the day i was pretty much born and fell in love with it. since i could speak to the day i was 11 i begged for a dog. and my parents finally gave in and got me one.
ways to prove it?
like she said, look after other dogs.
the biggest thing i ever did was, when i was bored i would say:
"Mom, if i had a dog i would always have someone to do something with and you wouldn't have to listen to me complain ever again that i'm bored."