If your mom is not willing to buy the bra, buy one yourself with your pocket money because if you have a bust and asking on here, you are probably old enough to go to the store by yourself?
For nipple teasing, it has nothing to do with your size, it happen regardless whether you are -A or G+ cup. A cropped tank, Bandeau, nipple pasties or sport bra can help, so does wearing something warmer or looser. If you don't have funds, perhaps you should try one of these methods instead?
Also, you should wear a bra if YOU are comfortable or want to wear one, not because you're classmate are wearing one. Not everyone wear a bra because of the stage the bust is developed. It's very individual.
I'm D so I need to wear one for health reasons (I.e my back would kill me if I don't) but I do experience the nipple thing, but if I don't feel like wearing one, I would probably wear something underneath my shirt or something warmer! I'm sensitive to cold so I get the nipple stand a lot.