Pat salmon fillets dry with paper towels and bring to room temperature if possible; season all over with the salt and pepper.
Squeeze about 1–2 teaspoons of the lemon juice over each fillet and rub it into the flesh.
Heat the olive oil in a large non-stick skillet over medium-high heat until hot but not smoking.
Place the fillets flesh-side down in the pan, pressing lightly so the surface contacts the pan; sear undisturbed for 3–4 minutes, until crispy and golden.
Flip the fillets and sear the other side for about 2 minutes.
Add the butter, chopped garlic, 3 tablespoons of the chopped parsley, the remaining lemon juice, and the lemon slices to the pan; spoon the melted butter and garlic over the fillets.
Cook for an additional 1–2 minutes, basting with the butter, until the salmon reaches your desired doneness and the butter begins to brown slightly.
Taste and adjust seasoning with salt, pepper, or more lemon juice if desired; garnish with the remaining parsley and drizzle the sauce over each fillet, then serve immediately.