If you were rear-ended by another vehicle that was traveling at 15-20 mph, the damage is definitely more than cosmetic, even if it might appear to be merely cosmetic to the naked eye.
The car needs to be put on a lift by a qualified body shop in order to determine just how many parts (there will be several…) need to be replaced in addition to the bumper cover.
Most body shops will give you a written estimate–gratis–in the hope of getting your business.