In a small bowl, combine the first six ingredients. Set aside 1/4 cup for serving.
Flatten chicken slightly. Cut red pepper in half lengthwise and remove seeds. Brush with oil and sprinkle with pepper. Grill chicken and red pepper, covered, over medium heat or broil 4 in. from the heat for 6-8 minutes on each side or until a thermometer reads 170°. Baste chicken occasionally with mayonnaise mixture.
Chop red pepper and toss with parsley. Serve chicken with red pepper and sauce mixture.