Here are a few questions you may want to ask your doctor before jumping:
Is the compression fracture centered? Or more of an anterior compression? If it was a posterior compression you would be having neurological symptoms, so that is probably not the case. If it is an anterior compression you are going to love to skydive. Extension (arching) feels great and the opening shock is a great traction to your back. If it is a central compression you may take longer to warm up before jumping and may need to return to the sport very gradually.
Ask you doctor about the disc height. How is it compared to the levels around it? I question if you really lost 1.75 inches in height. I don't doubt the recent height was taken accurately, but when was the 5'3' measured. When in high school? At the same time of the day? We are all taller in the AM when we first get up - before gravity gets us.
Ask you doctor if the nerve roots look clear and does he/she think any future problems will arise. My guess is you would know if you had any nerve root impingement, but the canal can be narrowed and you not have any symptoms.
I've had an anterior compression fracture in mid thoracic (at about the bra level - for you guys the place your back always itches and you can't reach). Skydiving is my therapy and medicine. It cures all aches and pains I may have! Good luck