Preheat broiler. Broil chicken on the unheated rack of a broiler pan 4 to 5 inches from heat for 10 minutes.
Lightly brush onion slices with olive oil. Remove broiler pan from oven. Turn and move chicken to one end of pan. Place onion slices in a single layer on opposite end of pan. Broil for 15 minutes or until chicken is no longer pink and juices run clear (180 degrees F).
Meanwhile, in a medium saucepan combine ketchup, molasses, vinegar, brown sugar, paprika, and pepper sauce. Bring to boiling over medium heat. Remove from heat; keep warm. Remove onions from broiler. Broil chicken 2 minutes more; brush with sauce during the last minute.
Chop two of the onion slices; stir into sauce. Serve chicken with onion slices and parsley.