Heat olive oil in a heavy skillet over high heat until it just begins to smoke. Place shrimp in an even layer on the bottom of the pan and cook for 1 minute without stirring. Season shrimp with salt; cook and stir until shrimp begin to turn pink, about 1 minute. Stir in garlic and red pepper flakes; cook and stir 1 minute. Stir in lemon juice, caper brine, 1 1/2 teaspoon cold butter, and half the parsley.
Cook until butter has melted, about 1 minute, then turn heat to low and stir in 1 1/2 tablespoon cold butter. Cook and stir until all butter has melted to form a thick sauce and shrimp are pink and opaque, about 2 to 3 minutes. Remove shrimp with a slotted spoon and transfer to a bowl; continue to cook butter sauce, adding water 1 teaspoon at a time if too thick, about 2 minutes. Season with salt to taste.
Serve shrimp topped with the pan sauce. Garnish with remaining flat-leaf parsley.