Make sure salmon filets are close to room temperature, and pat dry with plenty of paper towels. Season both sides with salt and pepper to taste. Drizzle the salmon filets with 1 tbsp lemon juice and rub into each.
Heat a non-stick, heavy bottomed pan over medium-high heat and add in olive oil. Swirl pan to evenly distribute the oil and once hot, place salmon filets flesh-side down and press lightly to get a good contact. Let the salmon sear for about 3 to 4 minutes undisturbed, to get a good crust on them.
Carefully flip salmon over and add butter and white wine to pan. Add in garlic and remaining lemon juice as well as 1 tbsp parsley. Spoon the butter sauce over the salmon and continue cooking until salmon is cooked through to your liking.
Before serving, top with lemon slices and spoon extra butter sauce over the top. Garnish with freshly chopped parsley. Enjoy right away.