Season the pork. Pat the pork shoulder dry with paper towels. Season with 1 tsp salt and 1/2 tsp black pepper.
Brown the pork. Heat a large Dutch oven or heavy pot over medium-high heat. Add 1 tbsp olive oil and brown half of the pork in a single layer until deeply golden, about 2–3 minutes per side. Transfer to a plate. Add the remaining 1 tbsp olive oil and brown the second batch. Transfer it to the same plate.
Cook the vegetables. Reduce the heat to medium. Add 2 tbsp butter, then stir in the onion, carrots, and celery. Cook for 6–7 minutes, stirring occasionally, until the vegetables begin to soften and pick up a little color.
Add the garlic and seasonings. Stir in the minced garlic, 1 tsp sweet paprika, and 1 tsp dried thyme. Cook for about 30 seconds, stirring, until fragrant.
Add the flour. Sprinkle 2 tbsp all-purpose flour over the vegetables. Stir well and cook for about 1 minute to remove the raw flour taste.
Stir in the tomatoes. Add the crushed tomatoes and cook for about 2 minutes, stirring occasionally, so the tomato flavor begins to deepen.
Add the broth. Gradually pour in 2 cups broth, stirring and scraping up the browned bits from the bottom of the pot. Stir in 1 tbsp Worcestershire sauce and add the 2 bay leaves.
Return the pork. Add the browned pork and any juices from the plate back to the pot. Stir well and bring everything to a gentle simmer.
Simmer the pork first. Reduce the heat to low, cover the pot, and simmer gently for 30 minutes. This gives the pork time to start becoming tender before the potatoes are added.
Add the potatoes. Stir in the potatoes, remaining 1 tsp salt, and remaining 1/4 tsp black pepper. Cover again and continue simmering gently for another 30 minutes, or until the potatoes are tender and the pork is fork-tender.
Finish the stew. Remove the lid and stir in 1 tsp sugar to soften the acidity of the tomatoes and bring the savory flavors into better balance. Simmer uncovered for another 5 minutes, until the sauce becomes slightly thicker and richer. Remove the bay leaves and stir in 3 tbsp chopped parsley.
Rest. Remove the pot from the heat and let the stew rest for about 10 minutes. This gives the sauce time to settle and thicken slightly.
Serve. Spoon the hot pork stew into bowls, making sure every serving gets plenty of tender pork, potatoes, vegetables, and rich tomato sauce. Finish with the remaining 1 tbsp fresh parsley and serve with crusty bread on the side.