Preheat grill for medium heat and lightly oil the grate.
Stir mayonnaise, Parmesan cheese, garlic, red pepper flakes, paprika, salt, and pepper together in a large bowl. Add potatoes and onion; stir to coat completely.
Arrange aluminum foil squares on a work surface and spray each with cooking spray. Divide potato mixture between the four squares. Fold aluminum foil over potato mixture and crimp edges together to form a secure pouch.
Cook on the preheated grill until potatoes are tender, about 30 minutes.