Pat the salmon fillets dry. Season both sides with 1/2 teaspoon salt and 1/8 teaspoon black pepper.
Grate the lemon to get 1 teaspoon zest and squeeze about 2 lemons to yield 4 tablespoons lemon juice; set aside.
Heat a large 10–12 inch light-colored skillet over medium heat. Add 4 tablespoons butter and swirl, stirring frequently, until the butter begins to turn light brown and solids become golden (about 3–7 minutes).
Carefully add the seasoned salmon to the hot butter, skinless side down. Cook uncovered 3–4 minutes until the first side is golden brown.
Flip the fillets and cook another 2–3 minutes until salmon is opaque and flakes easily, or reaches an internal temperature of 145°F.
In the last 2 minutes of cooking, add the grated lemon zest and 4 tablespoons lemon juice to the pan. Spoon the butter-lemon sauce over the fillets as they finish cooking.
Transfer salmon to plates, drizzle with the pan sauce, sprinkle with minced parsley and additional black pepper to taste, and serve immediately.