Crush the black peppercorns with a mortar and pestle until coarsely crushed; alternatively place them on a work surface and crush with the flat side of a knife or a heavy skillet. Set aside.
Place the all-purpose flour on a shallow plate. Pat the flounder fillets dry with paper towels, then season both sides with the ground black pepper and 1 teaspoon of the kosher salt. Dredge the fillets in the flour, shaking off any excess.
Heat the vegetable oil in a skillet over medium-high heat until shimmering. Add the floured fillets and cook for about 2 minutes on the first side, then gently flip and cook about 1 minute more, until the fish is opaque and flakes easily. Transfer the cooked fillets to a plate and set aside.
Wipe the skillet clean and return it to medium-high heat. Melt the butter, then add the crushed black peppercorns. Cook, stirring occasionally, for 2 to 3 minutes until the butter is fragrant and the pepper is toasted, taking care not to burn it.
Remove the skillet from heat and stir in the lemon zest, lemon juice, and the remaining 1 teaspoon kosher salt. Mix to combine.
To serve, spoon the buttery black pepper sauce over the flounder fillets and garnish with the chopped parsley.