1. Dry the salmon well with paper towels and season it all over with salt. In a small bowl, stir together the parsley, lemon juice, and 1 tablespoon of the olive oil. Coat the salmon fillets with the parsley mixture and set them aside at room temperature for 15 minutes.
2. Set a large skillet or nonstick frying pan over medium-high heat and let it get hot, about 2 minutes. Add the remaining 1 tablespoon olive oil and the butter. As soon as the butter is melted, add the salmon fillets, skin-side down. Reduce the heat to medium-low and cook, pressing down on the fillets 2 or 3 times with a spatula, until the skin is crisp, about 6 minutes.
3. Flip the salmon and cook until the flesh is done to your liking, the exact time will depend on the thickness of the fillets. If necessary, give it 1 or 2 minutes more, but be careful not to overcook. Transfer the fish to a warmed serving platter and cover loosely with aluminum foil while you make the sauce.
4. Increase the heat to medium and add the butter in a pan. When the butter melts, add the shallot and garlic and cook, stirring, until fragrant, about 2 minutes.
5. Add the wine, lemon zest, and lemon juice and bring to a boil until the sauce thickens slightly, about 2 to 3 minutes. Stir in the capers and season with black pepper. Pour the sauce over the fish and serve immediately.