Heat the butter in a large skillet over medium heat. Add the shallots and thyme and cook until the shallots are translucent, about 7 minutes; season with salt and pepper. Add the mushrooms and cook, stirring occasionally, until brown and tender, about 8 minutes. Discard the thyme sprigs.
Meanwhile, season the fish with salt and pepper. Heat the oil in another large skillet over medium-high heat. Add the fish, skin-side down, and cook until the skin is crisp, 3 to 4 minutes; transfer to a plate. Add the wine to the skillet and scrape up any browned bits with a wooden spoon. Boil over high heat until syrupy, 2 to 3 minutes. Add the cream and simmer to thicken slightly, 2 to 3 minutes. Season with salt and pepper.
Pour the cream sauce over the mushrooms in the skillet, then add the fish, skin-side up. Simmer until the fish is just firm and translucent, 5 to 7 minutes. Garnish with parsley, if desired.