It seems to me that he isn't letting you have it because he knows you want the fish so much and are putting up a fuss (fair enough though) He probably thinks he has power over you and if I had power over my sisters I wouldnt give it away that easily.
Hmm if I were you I would rationalize with your parents but try not to get emotional (I know this can be hard, but facts tend to lose their value when emotions are brought into play)
Say that him having the betta is pointless since it is teaching him poor life lessons, then throw in some examples of how he is learning that if he doesn't want to do something then someone else will always do it for him.
Then go in to say something about hes not learning to be responsible bla bla bla...main factor here is to be mature and emotionless.
Then do something to either
a) make him think you dont want the fish anymore...his power over you is now useless.
b) give him a very good reason to give you the fish....(eg "hey! lets try and breed it...then we can sell the babies and buy awesome things") but of course the fish needs to be in your larger tank and you need to condition for a month (he doesn't know thats not the case
) then procrastinate on getting the "right female" because that will give you the "most expensive babies" by then he would have forgotten and you will have a new fish in your tank.