Combine the flour, slat, and pepper and dredge the short ribs. Heat half the olive oil in a Dutch oven over medium-high heat, then sear the beef on all sides until golden brown, 3 to 4 minutes per side. Remove the beef and set aside.
Add the onion and garlic and cook until brown and fragrant, about 2 minutes more.
Add the stock, scraping up any bits stuck to the pan. Return the beef to the pot. Bring to a boil. Cover pot and transfer to the oven. Cook until the beef is fork-tender, 2-1/2 to 3 hours.
In the last 15 minutes of cooking, heat the remaining oil in a heavy skillet over medium heat. Add the mushroons and cook until golden, 6 to 8 minutes. Toss with the parsley and season with salt and pepper.
Remove the ribs to serving dishes, place the pot over high heat, and reduce the cooking liquid until slightly thickened. Serve with the mushrooms and cooking liquid.