In a bowl, mix salt and sugar. Add shrimp and stir gently to coat. Cover and chill 45 minutes to 1 hour. Rinse shrimp well and drain; also rinse and dry bowl.
Return shrimp to bowl. Add olive oil, parsley, lemon peel, garlic, and pepper. Mix to coat. Thread shrimp on metal or soaked wooden skewers, running skewer through the body once near the tail and once near the head end of each shrimp so it looks like the letter C.
Lay shrimp skewers on an oiled barbecue grill over hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook, turning once, until shrimp are bright pink and opaque but still moist-looking in center of thickest part (cut to test), 5 to 6 minutes total. Serve with lemon wedges to squeeze over shrimp.
Wine Pick: Dry rosé or Sauvignon Blanc. The shrimp are both sweet and salty, so they call for some soft, round fruit balanced with crispness in a wine. Bonny Doon Vineyards' Ca' del Solo Big House Pink ($10) meshes perfectly and is a great toes-in-the-sand beverage; Joel Gott's Napa Valley Three Ranches Sauvignon Blanc ($18) works well too.