That's just wrong that your parents won't let you use deodorant. Maybe try mixing a little water with some baking soda, and applying it to your underarms. Not too wet, so it will dry alright, hopefully. That may work.
First thing would be washing with soap and water. And don't tell me you parents won't let you shower or bathe. If you have to, do it a couple of times a day. If you're at school make sure you shower after gym class, a wash your arm pits really well. I'm not going to pry into your family religion, or anything ;like that, but what is their reasoning for no deodorant? Tell them, it is part of proper hygiene, and necessary in normal society.
Try talking to them, 13 is the age that kids need to start wearing an anti-perspirant deodorant. Make sure it is an anti-perspirent. You need to make your parents understand that they are not being reasonable, but say it nicely.
Wash with soap and water before leaving home. Also how about using an anti-perspirant? They are different you know. Some products contain both deodorants and anti-perspirants and others are one or the other. One example is Mitchum non scented anti-perspirant. This is NOT a deodorant. It PREVENTS odor. Maybe your parents aren't this technical.
I don't know your age,but if you haven't got much growth in your pits,just a warm,soapy washing and rinsing twice each day ought to do it. if you've gotten a little hairy,start with a scented talc,if your parents won't hang with that,just use plain unscented baby talc. won't allow deodorant?!!? neveh hoid a sucha thingg!