Do you know anyone with rabbits? You could get the hay from someone or maybe even a farm instead of buying a bag of it just for testing purposes. Also, there are many different kinds of hay so if your mom is indeed allergic to timothy hay, other grass hay options you could eventually try out are orchard grass, oat hay etc. The dust in hay could also trigger allergies not not necessarily the hay type itself.
Domestic rabbits are unlike wild rabbits and do not adapt well to the outdoors. They are extremely fragile and sensitive animals, so the noise level outdoors and other elements like the weather (thunderstorms etc) can easily make them scared or even go into gut stasis, which can be fatal. If you are still under your parents’ care, it is extremely important that they understand and agree to the commitment required to have a bunny. It’s not easy and can also be expensive.
Rabbits don’t smell; only their pee does. If you clean their living quarters regularly you wouldn’t be able to smell anything except maybe the smell of hay. I recommend reading the BUNNY INFO section to learn more and even get your mom to read it if you can 