Gently combine ground bison with parsley, 2 tablespoons hot pepper jelly, and salt. Form mixture into 4 equal patties.
Heat a lightly oiled panini press or grill pan over medium heat, and cook burgers, pressing them lightly (if using a grill pan, turn them halfway through) about 2-3 minutes. (Use a meat thermometer to check that burgers have reached 160.)
Transfer to a plate.
Place burgers on bottom sides of buns, and spread each with 1 tablespoon hot pepper jelly. Top with onion and a slice of cheese. Top with other half of bun, and place back on panini press or grill pan with a heavy lid on top. Press about 3 minutes total (turning halfway if using a grill pan) for medium or until bread is golden brown and burger is cooked to desired doneness.
