Let turkey breast stand at room temperature 30 minutes; rinse with cold water, and pat turkey dry.
Stir together parsley, next 4 ingredients, and 1 tsp. salt. Loosen and lift skin from turkey without totally detaching skin, and rub parsley mixture under skin. Replace skin. Sprinkle cavity with 1/2 tsp. salt; rub into cavity. Sprinkle remaining 1/2 tsp. salt on skin; rub into skin.
Light 1 side of grill, heating to 350° to 400° (medium-high) heat; leave other side unlit. Place turkey over lit side, and grill, without grill lid, 4 minutes on each side or until golden brown. Transfer turkey to unlit side, and grill, covered with grill lid, 2 to 2 1/2 hours or until a meat thermometer inserted into thickest portion registers 165°. Remove from heat, and let stand 30 minutes before slicing.