Transfer the cooked fillets to a warm platter. Add another 2 tablespoons unsalted butter to the skillet if needed and repeat with the remaining fillets, 3–5 minutes total. Transfer to the platter.
Pour off any dark fat and quickly wipe the skillet. Return the skillet to medium heat and add the remaining 6 tablespoons unsalted butter. Cook, swirling, until the butter foams, then turns deep hazelnut brown with a nutty aroma, 2–3 minutes.
Remove the skillet from heat and immediately stir in the fresh lemon juice and fresh parsley. Swirl to combine, then spoon the hot brown-butter sauce over the fish.