Make rémoulade: Combine mayonnaise, mustard, pickle, lemon juice, scallion, Tabasco sauce, paprika and parsley in a bowl. Cover with plastic wrap and refrigerate.
Heat gas grill to medium. Thread shrimp on metal skewers, brush with oil and sprinkle with salt. Grill shrimp until cooked through, about 2 minutes per side. (Or place on a foil-lined baking sheet and broil 6 inches from heat until cooked through, 2 minutes per side.)
Remove shrimp from skewers and serve with rémoulade on the side.