Preheat oven to 350°. Pour vegetable oil into a deep medium pot to a depth of 1" and heat over medium heat to 325° on a candy thermometer. Pick leaves from 2 sprigs of the parsley and fry in oil until crisp, about 30 seconds. Drain on paper towels and set aside. Cut fish filets in half crosswise with a sharp knife, then score skin in a crosshatch pattern and set aside. Chop fish bones and head into large pieces and set aside.
Heat 2 tbsp. of the olive oil in a heavy-bottomed medium roasting pan on top of stove over medium-high heat. Add onions, carrots, celery, bay leaves, and remaining 6 sprigs parsley and cook, stirring often, until just softened, 3-5 minutes. Add fish bones and head and cook, stirring often, until bones begin to turn golden brown, 3-4 minutes. Add wine and tomato paste and cook, scraping browned bits stuck to bottom of pan, until alcohol evaporates, about 2 minutes. Whisk flour and 2 cups water together in a bowl until smooth, then stir into pan and bring to a boil. Transfer pan to oven and roast until liquid has reduced by about half, 25-30 minutes.
Strain liquid through a large sieve into a medium bowl, pressing on solids with back of spoon (you should have about 1 cup liquid). Discard solids. Set aside 1⁄4 cup of the liquid and put the remaining 3⁄4 cup liquid into jar of blender. With motor running, gradually add 2 tbsp. of the olive oil through hole in top of blender lid, pureeing until sauce is emulsified. Transfer sauce to a small saucepan, season to taste with salt and pepper, and keep warm over lowest heat.
Heat 1 tbsp. of the olive oil in a medium skillet over medium heat. Add mushrooms and cook, stirring occasionally, until they begin to brown, 3-5 minutes. Add reserved 1⁄4 cup liquid and season to taste with salt and pepper. Remove skillet from heat, cover, and set aside.
Heat 2 tbsp. of the olive oil in a large skillet over high heat. Season fish filets with salt and pepper to taste and sear in skillet, skin side down, until skin is golden and crisp, 1-1 1⁄2 minutes. Turn filets, reduce heat to medium, and cook until fish is just cooked through, 2-5 minutes.
To serve, divide fish between two warm plates, placing them skin side up. Top fish with mushrooms, then spoon sauce over mushrooms and around fish. Drizzle each plate with some of the remaining 1 tbsp. olive oil and garnish with fried parsley.