I haven't read any where that it says pet allergies are from urine but I did find that allergies are from dander and saliva. The best bet would be to keep the dog out of your room if you can and replace the bedding and/or flip the mattress over if you can't afford a new one. I know it's tough when you have a parent that insists they are always right but maybe you can show her this article from the Mayo clinic. http://www.mayoclinic.com/health/pet-allergies/HQ01204
If you've been tested and it showed positive for animal allergies maybe you can show your mom the results to have her believe you.