In a large (10- to 12-inch) cast-iron skillet, melt the butter over medium-high heat. Season the trout with 1/2 tsp. each salt and pepper. Place the trout, skin side down, in the skillet. Scatter with the thyme and lemon slices.
Cook the trout, spooning the butter from the bottom of the skillet over the fish often, until just cooked through (do not flip), about 5 minutes. Transfer the fish and lemon slices to a serving plate.
Remove the skillet from the heat and stir in the lemon juice, scraping up any browned bits from the bottom of the pan. Spoon the pan sauce over the fish. Top with the parsley.