Welcome to the cavy world!
I first would just realize that if you rely on your mom for the money to care for the pigs and she doesn't want any (regardless of what you tell her) then you have to accept that and wait till you have your own place. That said, you could just take your time to understand as much as you can about their care. Read the care guide at Guinealynx.info and check out their medical one too. Read the stickies on these forums and just read random topics.
Make a little spreadsheet that goes over the cost of all the items you need to get started out, their monthly cost, and vet care (which can be around a thousand dollar per pig, worst case scenario). Once you itemized how much keeping the guinea pigs will cost (proper cage, proper food, scale, vegetables etc.) and made a schedule/list of care tasks you will do every day, once a week etc. then you can bring this information to your mom to show you've done your research and to give her a realistic cost analysis. Don't try to trick her and make it seem less expensive than it is. That honesty and effort put into it will hopefully show your mom that you're being mature and serious about owning Guinea Pigs.
While you do that, depending on your age, try and see if you can get a part time job some place and save up money. This isn't just a good routine to get into for your own future but will show you're willing to contribute to the cost of these pets and you're willing to take some responsibility for them. If you're too young or you simply can't at this stage, start taking up chores in the house you didn't do before and stick to them, without being asked. This shows you'll be mature enough to look after these pets day in, day out, regardless of day.
Don't expect your mom to take over unless you're seriously ill. Do talk to her and everyone in the household to make sure they are okay with these pigs being there and in case of emergency they are willing to take care of them for a few days. You might also want to make sure nobody is allergic to the pigs or their hay. They are also quite loud and preferably you want to keep them in an active area of the house like the living room or your bedroom if you are there often. So make sure people are okay with that arrangement.
Also think about your future, where are you going to be in several years from now? College dorms don't necessarily allow pets, who would look after them? Guinea pigs can live as long as 9-10 years so make sure they fit into your future plans and that you have a back up in case something goes wrong.
Anyway, this is all super serious but I do believe the above is the best way to show a parent you are mature and responsible enough to look after these pets. Especially if you expect your parents to pay for them. They do require a lot of care and they can get very expensive to own so make absolutely sure they are right for you. That said, I can tell you one thing, if you do decide to adopt a couple of pigs. They are the best little creatures ever and bring so much joy and happiness, amazing pets and well worth the expensive and effort. Just make sure you're ready to care for them fully. They are not easy/cheap/starter type pets at all.
Good luck and I hope you can find your way to join the Guinea Pig Slavedom
