The goal shouldn't be to justify the bike, it should be to prove you are respondsible enough to have one. If you are known to do stupid things, I wouldn't count on them agreeing with your decision to get one.
I was fortunate enough to have 2 parents that used to ride so I started on dirtbikes when I was young. By the time I was 16, I was already capable on 2 wheels so my parents didn't have to worry about me learning to ride while dogging cages. My father was an easy sell to get a bike. He knew if I got one, that would be his excuse to start riding again. That is also the reason I ended up with a much larger starter bike than I had originally wanted. It scared the shit out of me at first, but now I'm glad I have it.
I honestly thought my mother would be opposed to it because she has worked in hospitals for years and gets to see first hand what happens in motorcycle accidents. She let me get it though, without any kind of fight because she trusts that I won't do anything stupid on it and I haven't let her down yet. (It also helps that I don't tell her about some of the things I do when I ride).
Your parents most likely know that the bike is inherently more dangerous than a car, so figure out how to sell yourself to your parents!
I prefer to step lightly and carry a BIG stick...