Definitely sounds like you have your heart set on an AR, so far be it from me to tell you not to. I don't know how much this will help but here are my thoughts.
I bought my wife an AR-15 when she was 20. She loved it. flash forward 7 yrs. It hardly ever gets shot. Unless you have a specific use in mind (hunting, target shooting, competition, etc.) you may find that after a while, the coolness of owning an AR wears off. Not saying it will, just a thought.
Without knowing you personally, I'd say you seem pretty responsible and I don't see why you should not have one if you want one and the law allows it.
I agree with a lot of the posters on being cool and mature while discussing this with your parents. If your Dad has not put his foot down and said no to an AR, then it seems like he may be won over with logic. Study up on ARs extensively and share that knowledge with him. As far as the "Assault Rifle" issue, yuo can buy 25 rd mags for a 10/22 as well as buying lower capacity (10 or 20 rd) mags for an AR.
Check and double check how your local laws apply!!!
We have a Bushmaster with a 16" barrel and full handguard. Other than the shorter portion of exposed barrel, it looks in all respects like a military issued M16A2 (minus the 3 rd burst, of course). I have not looked into ARs in a while, but when I did, it seemed like the fewer bells and whistles, and the more "A2" it looked, the lower the prices tended to be. I can't imagine that you couldn't find a decent AR + .22 conversion for the money you have saved, or maybe just a shade more (probably no more than you could save in the mean time, while you continue AR research).
As with all firearms, follow all of the safety rules and be generally responsible with all of your weapons and I can't see how this could a bad purchase or "worse" than a.22.
Good luck and let us know what you do.