Prepare grill for direct heat cooking. or position broiler rack so that top of shrimp on rack in pan will be 4" from heat source; preheat broiler. In food processor or blender, puree peppers, horseradish sauce and mayo until smooth; reserve.
With small, clean scissors, cut through shell at back of each shrimp.
Gently peel away shell, leaving tails on. With small knife, loosen dark vein from center; remove. Rinse shrimp; pat dry.
In bowl, toss shrimp with oil and Old Bay seasoning. Place on grill or rack set in broiler pan. Grill or broil, turning one, until pink and opaque, 2-3 minutes per side. Transfer to platter. Transfer reserved pepper mixture to serving bowl or, if desired, to pepper. Sprinkle shrimp and dip with parsley.