Pat the trout completely dry, season both sides with the salt and pepper and dust only the skin sides lightly with the flour.
Heat the oil in a large nonstick skillet over medium-high heat until shimmering.
Lay the fillets skin-side down and press them flat with a spatula for 20 seconds so the skin does not curl.
Cook without moving until the skin is crisp and the fish is opaque two-thirds of the way up, 3 to 4 minutes.
Turn the fillets and cook until the centers reach 145°F and flake under gentle pressure, 1 to 2 minutes; transfer them to plates.
Lower the heat, melt the butter in the skillet until foamy and stir in the lemon juice and parsley for 20 seconds.
Spoon the lemon butter over the trout and serve with the lemon wedges.