Rinse pork neck bones well and pat dry. Parboil for 5 minutes if needed to remove impurities.
Heat olive oil in a pot and brown neck bones on all sides. Set aside.
Add onions, celery, and carrots. Sauté 4–5 minutes until softened.
Stir in garlic, smoked paprika, cayenne, pepper, and thyme; cook 1 minute.
Add neck bones back in with broth, tomatoes, bay leaf, and Worcestershire.
Bring to a simmer, lower heat, cover, and cook 1.5–2 hours.
Add potatoes; simmer 25–30 more minutes.
Adjust salt, remove bay leaf, pull off meat, discard bones.
Serve hot with parsley or green onions.