Ok, let me start with some (not all) of the folks that have answered - are ya'll just crazy or what? You can't just go out and get another without permission - it's the parent's house, not the questioner. Perhaps, as one said, the parents are still upset about the loss of the 20 yr old cat, and are not ready to "replace" it yet. Also, that cat lived 20 years, are you ready or your parents to give that kind of commitment to another animal, or will you be leaving for college, marriage....and leave the cat with them. Then, you have to respect their wishes. Pictures of kittens are all cute (never saw one that didn't melt my heart) - but the responsiblity of a kitten is great. You must teach him/her everything, not to mention the cost of new kitten vet care, litter, food, toys and everything else. Talk to your parents, tell them why you would like a new cat (not just that it's cute and here's a picture...). If you have a job, offer to pay "rent" for the kitten to cover any cost (they can put it in a .... pardon the pun....a kitty fund) - and assure them, when you leave, the kitty will go with you if THEY wish. Perhaps they might, if they decide to get it, want to keep it with any companion animals they may already have.
My son wants to come home and take "his" cat to Alaska (I live in SC)- even though he's in the military and on deployment often - and I told him absolutely not. The cat has been in our care since he was born, and have his buddies he lives with. I explained it would be cruel to separate an older cat from his buddies, and if he wants a cat that bad, he should go to a local shelter and adopt, but be prepared to have a cat-sitter or back-up owner ready in case he has to move overseas (which is likely).