At 14, you shouldn't have to be responsible for the cost of caring for the animal. Instead, talk to your parents about being willing to do the everyday things for the cat, like cleaning the litter box. Also, offer to do extra chores around the house and keep your grades up, that sort of thing.
(the picture of Boo that you posted is adorable
While this can be very cheap, you don't want to feed the cat the least expensive food you find. I would rather you feed cheap food than re-home the cat though, just because so many cats need homes. If you live by a Costco, the Kirkland Signature dry food is not bad. It's manufactured by Diamond Pet Foods. You can get a huge bag (something like 40 lbs. I think, it would last several months) for I think only $15. It's better and less expensive than most grocery store pet food brands, at least as far as I can remember. You'll want to feed your cat wet food, which ranges from $0.30 - $2.00 a can, varying widely in quality. I spend a minimum of $45/month on mostly premium wet food for 2 cats. For one cat's wet food each month on a tight budget, I would set aside $10-15.Litter
This can be very cheap if you use pellets from a feed store (I believe this would be less than $5/month). If you want clay clumping litter, it'll be maybe $20/month?Vet Care
This is the tough part. Keeping the cat indoors will keep vet bills down immensely, plus you won't have to buy flea/worm medication. For spaying/neutering, a low-cost clinic will be able to do it for not too much. The one by me charges $45 for females and $28 for males. Micro-chipping they do for $15. Most vets will do the initial first check exam for free. Vaccines can be done usually at low-cost clinics also. This is basic vet care, if your cat has any health problems that arise and you want to keep the cat, your parents must be willing to take responsibility for them.
I hope the kitty can stay with your family