Preheat the oven to 325°F. Place the pork on a cutting board, pat dry with paper towels, and season with the salt and pepper.
Add the celery, carrot, and onion to the pan and cook, stirring often, until soft, 5 to 7 minutes. Stir in the garlic and cook until fragrant, about 2 minutes. Stir in the tomato paste and cook until it starts to turn a little darker, about 3 minutes. Sprinkle with the flour and cook another 2 minutes, stirring constantly. Whisk in the wine, beating out any flour lumps. Simmer the wine until it is reduced by half, 2 to 4 minutes.
Bring the liquid to a simmer, turn off the heat, cover the pot, and place the pot in the oven to braise until the meat shreds easily when pulled by a fork, about 3 hours. Transfer the pork to a platter, taste, and season with more salt and pepper if needed. Remove and discard the bay leaves and parsley stems. Serve the sauce on the side.