Pat salmon dry with paper towel. Season both sides of each piece of salmon with salt, and black pepper if you're using it.
Heat a cast iron skillet over medium heat. When hot, add olive oil. Place salmon pieces in skillet with skin side down. Press down gently on the salmon pieces to make sure the surfaces of the fish have good contact with the pan, but do not move the fish around in the pan. Allow the fish to cook for about 3 minutes until the skin is crisp, then flip the fish over and cook the second side 2-3 minutes, depending on thickness of fish pieces. Remove the cooked fish to a plate.
Add white wine to the skillet. With a wooden spoon, scrape the bottom of the pan to loosen any browned bits. Simmer wine until reduced by half, about 2 minutes.
Add garlic and cook until garlic is translucent and the mixture is fragrant.
Stir in chicken stock, lemon zest, lemon juice, and capers and cook until the sauce has thickened, 4 to 6 minutes.
Add butter to pan, whisk vigorously until sauce has thickened, about 1 minute.
Return the salmon to the skillet, flipping gently a few times to coat the pieces on all sides with the sauce. Spoon the sauce along with capers on top of each piece of salmon. The salmon is cooked when a thermometer inserted into the middle of the thickest fillet should register 120 to 130°F for medium-rare, or 135 to 145°F for medium-well.
Transfer cooked salmon to plates, along with some of the sauce and capers. Garnish with lemon slices and chopped parsley.