Heat gas or charcoal grill. Cut 18x12-inch sheet of heavy-duty foil; place on large cookie sheet. Spray foil with cooking spray. Unroll dough onto foil. Starting at center, press out dough into 14x10-inch rectangle. Spray dough with cooking spray.
Turn dough rectangle over directly onto grill rack; peel off foil, and return foil to cookie sheet, sprayed side up. Cook uncovered over medium-low heat 2 to 4 minutes or until bottom is golden brown. (Watch carefully to prevent burning.) With wide metal pancake turner, turn crust over. Cook 1 to 2 minutes longer or until bottom is set but not brown. Carefully remove from grill; place browned side up on foil. Close grill.
In medium bowl, place chicken and barbecue sauce. Using 2 forks, toss until chicken is coated with sauce. Top crust evenly with chicken mixture. Sprinkle with cheese; top with red onion. Slide pizza from foil onto grill. Cover grill; cook 3 to 5 minutes longer or until crust is golden brown and cheese is melted. Sprinkle with parsley. To serve, cut into 6 rows by 3 rows.