Preheat oven to 425°F. Line a rimmed baking sheet with parchment and brush with 1 teaspoon olive oil.
Pat shrimp very dry. Toss with mayonnaise, garlic, lemon zest, salt, pepper, paprika, and red pepper flakes.
Combine panko, Parmesan, parsley, and 2 tablespoons olive oil in a bowl until crumbs are evenly moistened.
Press each shrimp into the crumb mixture to coat both sides. Arrange on the prepared sheet in one layer.
Bake 10–12 minutes until shrimp are pink and opaque and the topping is golden.
Broil 1–2 minutes for extra crunch, watching closely to prevent burning.
Rest 2 minutes. Serve hot with fresh parsley and lemon on the side, if you like.