Pat dry the room-temperature salmon fillets with paper towels. Season all over with salt and pepper.
Heat the olive oil in a large non-stick skillet over medium-high heat. Place the salmon fillets flesh side down, pressing them lightly to ensure the entire surface of the flesh is in contact with the pan. Sear undisturbed for 5-6 minutes until crispy and golden.
Flip the fillets and sear the other side for 2 minutes. Remove the salmon to a plate.
In the same skillet, add the butter, chopped garlic, 3 tablespoons of parsley, the juice of 1 lemon, and the lemon slices. Place the salmon back in the pan and stir the butter and garlic around each fillet.
Continue to cook the salmon for an additional 1-2 minutes, or until it reaches the desired doneness (the butter will begin to brown slightly). Taste and season with salt, pepper, and more lemon juice if desired.
Garnish with the remaining parsley and drizzle the butter sauce over each fillet.