Pat the salmon fillets dry thoroughly with paper towels on all sides so they sear and brown instead of steaming.
Season the salmon on both sides with the sea salt, black pepper, and paprika, gently pressing the seasonings onto the flesh.
Heat the olive oil in a large skillet over medium to medium high heat until hot and shimmering but not smoking.
Place the salmon fillets skin side down in the hot skillet and cook without moving them until the skin is crisp and the salmon is mostly opaque, about 5 to 6 minutes depending on thickness.
Carefully flip the salmon and cook on the second side just until the center is barely opaque and flakes easily with a fork, about 2 to 3 minutes more, then transfer the salmon to a plate and tent loosely with foil.
Reduce the heat to medium low and add the butter to the same skillet, letting it melt and cook until lightly foamy and just turning a light golden color around the edges for a slightly nutty browned butter flavor.
Add the minced garlic to the melted butter and cook, stirring frequently, until fragrant and softened without browning, about 30 to 60 seconds.
Stir in the lemon juice, lemon zest, parsley, and crushed red pepper flakes, then add the water or broth and simmer briefly until the sauce is smooth and slightly thickened, about 1 to 2 minutes.
Taste the lemon garlic butter sauce and adjust with additional salt, pepper, or lemon juice to balance the richness to your preference.
Return the salmon fillets to the skillet, spoon the hot lemon garlic butter sauce over the tops, and warm gently for 1 to 2 minutes so the flavors meld without overcooking the fish.
Serve the salmon fillets hot, drizzled generously with extra lemon garlic butter sauce and garnished with additional parsley and lemon wedges if desired.