Prep the salmon. Pat salmon dry with paper towels. Season both sides with salt and pepper. Drizzle 1 tablespoon lemon juice over the filets; let sit 5 minutes while you heat the pan.
Heat the skillet. In a large nonstick or cast‑iron skillet, heat olive oil over medium‑high until shimmering.
Sear side one. Place salmon filets presentation‑side down. Cook 3–4 minutes, undisturbed, until deeply golden and the fish releases easily.
Flip and finish. Turn the filets and cook 3–4 minutes more, until the center reaches 125°F for medium (or to your preferred doneness). Transfer to a plate.
Build the sauce. Lower heat to medium. Add butter; when melted and foamy, stir in garlic. Cook 30–45 seconds until fragrant (don’t brown it).
Brighten and herb it. Stir in remaining lemon juice and 2 tablespoons parsley. Simmer 1–2 minutes to slightly thicken.
Coat the salmon. Return salmon to the pan; spoon sauce over filets. Tuck in the lemon slices. Warm 1 minute.
Serve. Plate salmon, spoon over extra sauce, and sprinkle with remaining parsley.