I got a 10/22 when I turned 13. I pawned it when I was a dumb broke college student. I have since bought another and it is a fantastic firearm. I have a couple dozen guns. The nice thing about the Ruger is that it is as accurate as most shooters, it is fun, reliable and there are tons of add ons - oh and they are cheap. It can grow with you and they are a lot of fun to work on. Mine currently has a Hogue overmolded olive drab stock with a Tactical Solutions flutted bull barrel, matte black with an olive drab compensator. It has a Leupold Var XII 3x9 (an extra as my big hunters now wear Zeiss). I plan on upgrading the trigger. Don't be alarmed though, straight out of the box they are a great tool. Lastly, every Walmart has them or can get them for you. Also pawn shops usually have several. Check your Walmart prices first.
As for your parents, if they shoot it is a no brainer, they should be happy to have a shared interest. If not, try to get one interested and make it quality time as you learn together.
Don't let anyone kid you, semi autos are more fun especially in 22 where the ammo is cheap and the targets are plentiful. I actually set up a 22lr "action" rifle and pistol course at the farm in Alabama when it isn't hunting season.
Good luck,
tjg