I haven't read the entire thread... GUILTY! But.. regardless, I am keeping my advice up here for an other youngters or adults that are thinking about getting a bird.
I am not sure how old you are, but is there a location in which you could volunteer to help birds or other animals? Are children permitted to do such a thing?
You get my line of thinking? You would get a chance to work with animals or birds in real life and have the experience needed, and they would have to spend the time driving you back and forth instead of having a green cheek already at home to take care of. Yes, you have another bird, but working with other parrots if you can can't hurt! Just be careful to wash hands, etc. so nothing would be spread from the birds you help to your pet.
After a little while I would think they would give in, or you would see how much goes into caring for these intellingent creatures and possibly change your mind.
Either way, you would get to know some parrots and get experience caring for them without the commitment of having your own. I know I would love to have one of my own, but having cared for my parent's parrot I know that my husband and I aren't home enough to give the attention a bird needs. We visit and parrot sit as much as we can instead.
Even if there is no place to volunteer it couldn't hurt, while you are waiting on your parents to warm up to your idea, to do research on things that you could do to help your feathered friends. Education, raising money, join a club for parrot lovers, etc.