In a bowl, spread flour. Line a rimmed baking sheet with parchment paper. Season fish fillets all over with salt and pepper. Dredge fillets in flour, shaking off excess. Transfer to prepared baking sheet.

In a skillet, heat oil until shimmering. Working in batches, add fish, presentation side down, and cook until well browned on on first side, 3-4 minutes. Carefully flip fillet and cook until just cooked through and flesh is opaque throughout, (30 seconds to 1 min). Transfer fish, to a serving platter.

Boil vinegar, add capers, unsalted butter, and butter-flour mixture and cook, whisking constantly, until butter has melted, and all emulsified nicely.

Whisk in lemon juice and parsley. Season with salt, then pour sauce over fish fillets and serve right away.