In a wide shallow bowl, combine all ingredients from olive oil through black pepper.
Add shrimp to bowl and fold gently to coat shrimp evenly with the marinade. Cover and refrigerate for 2 to 3 hours, folding gently about halfway through.
Remove shrimp from fridge. Then heat grill to medium-high. Make sure grates are clean, and then oil them well.
When grill is hot, set shrimp on grates and grill for 3 to 4 minutes per side for jumbo sized shrimp, using a tongs to flip. Cook just until shrimp are no longer translucent and have some nice char. Do not overcook the shrimp, as that will cause them to become tough and dry.
Place shrimp on a platter and sprinkle with parsley. Serve with fresh lemon wedges to squeeze over the top, plus cocktail sauce and/or tartar sauce for dipping.
