Preheat oven to 450 degrees. On a rimmed baking sheet, toss mushrooms with 3 tablespoons oil and thyme leaves; season with salt and pepper. Roast until mushrooms are lightly browned, 15 minutes.
Rub fish with 1 tablespoon oil; season with salt and pepper. Arrange fish over mushrooms; top with thyme sprigs. Roast until fish is opaque throughout, 10 to 12 minutes. Toss parsley with lemon juice and 1 teaspoon oil; season with salt and pepper. Serve fish and mushrooms with parsley salad and lemon wedges.