Preheat oven to 325°F with a rack in the center.
Pat ribs dry, season all over with salt and pepper, then dust lightly with flour.
Heat olive oil in a large Dutch oven over medium-high. Brown ribs on all sides, 8–10 minutes total. Transfer to a plate.
Lower heat to medium. Add onion, carrots, and celery. Cook, stirring, until softened and lightly browned, 5–7 minutes.
Stir in garlic and tomato paste. Cook until the paste deepens in color, about 2 minutes.
Pour in red wine and scrape up browned bits. Simmer 2–3 minutes to reduce slightly.
Add beef broth, Worcestershire, balsamic, thyme, rosemary, and smoked paprika. Return ribs and any juices to the pot; bring to a gentle simmer.
Cover and braise in the oven until fork-tender, about 2 hours 30 minutes.
Skim excess fat. Set pot over medium heat and simmer uncovered 10–15 minutes to thicken. Adjust salt and pepper.
Serve ribs with the sauce over mashed potatoes or polenta. Sprinkle with parsley.