If using a charcoal grill, light a chimney starter filled with charcoal (about 6 quarts) and allow to burn until coals are partially covered with ash, about 20 minutes. Open the bottom vent on the grill. Build a modified two-level fire by arranging coals over half the grill, leaving the other half empty. Set the cooking grate in place, cover grill, and heat until hot, about 5 minutes. If using a gas grill, turn burners to high to heat grill with lid down until very hot, about 15 minutes.

Use grill brush to scrape grates clean. Dip wad of paper towels in vegetable oil; holding wad with tongs, oil the grates.

Place the sauce ingredients (butter, lemon juice, garlic, pepper flake, and salt) in a 10-inch disposable aluminum pan or pie plate. Cook over the hot side of the grill, stirring occasionally, until the butter melt, about 1 ½ minutes; transfer to the cooler side of the grill (if using a charcoal grill) or off heat (if using a gas grill).

Place the shrimp skewers, sugared sides down, on the hot side of the grate and use tongs to push the shrimp together on the skewers if they have separated. Grill the shrimp, uncovered, until lightly charred, 4 to 5 minutes.

Using tongs, flip and grill until the second side is pink and slightly translucent, 1 to 2 minutes longer.

Using a potholder or oven mitt, carefully lift each skewer from the grill and use tongs to slide the shrimp off the skewers into the pan with the sauce. Toss the shrimp and sauce to combine and transfer the pan to the hot side of the grill; cook, stirring, until the shrimp are opaque and fully cooked, about 30 seconds. Remove from the grill, add the fresh parsley, and toss to combine. Transfer to a serving platter and serve immediately with lemon wedges.