If your dog is living a poor quality of life due to being paralyzed, it may be in the best interests of your dog to be put down.
Your mom may feel you have too much on your plate to take care of a cat as well as the dog and whatever else you have going on. Pets are also expensive and she may not be able to afford you having a cat as well. Talk to her about why you want to have a cat and ask her (nicely) why she thinks you should not have one right now. If you speak to your mother reasonably you are more likely to get the answers you are looking for from her.