Combine the onion, carrot, celery and garlic in a food processor and pulse until finely chopped.
Heat the oil in a Dutch oven over medium-high heat. Sprinkle salt and pepper all over the short ribs, add them to the pot and brown well on all sides, about 3 minutes per side. Transfer to a plate.
Add the chopped vegetables to the pot and cook, stirring, until slightly softened, about 4 minutes. Add the tomato paste and cook, stirring, 1 minutes. Add the tomatoes, wine and rosemary and stir to combine. Return the short ribs to the pot and bring to a simmer over medium-low heat. Cover the pot and cook until the meat separates easily from the bones, is extremely tender and shreds easily, about 3 hours. Remove the meat from the ribs and discard the bones. Shred the meat into small pieces, discarding any hard sinew and unrendered fat. Return the meat to the sauce.
Meanwhile, bring a large pot of salted water to a boil. Cook the linguine according to the package instructions.
Serve the pasta topped with short rib sauce, chopped parsley and Parmesan.