Place meat in a large bowl and cover with cold water. Add vinegar, cover bowl and refrigerate for at least 4 hours or overnight. Drain; pat dry.

Place a rack in the lower third of oven and preheat to 325ºF. In a large ziplock bag, mix flour, mustard, 1 tsp. salt and 1/2 tsp. pepper. Warm 2 Tbsp. oil in a large pot over medium-high heat. Working in batches, toss oxtail pieces in flour mixture, shake off excess and brown on all sides in pot, turning with tongs. Do not overcrowd the pan. Remove to a plate; repeat with remaining pieces, adding more oil to pan as needed.

Pour wine into pan and cook for 3 minutes, stirring to scrape up browned bits from bottom. Add onions and garlic and cook for 2 minutes, stirring occasionally. Stir in tomato paste; cook for 1 minute. Return meat to pan and add carrots, celery, leeks, crushed tomatoes, bay leaves and thyme. Season with salt and pepper. Pour in broth and bring to a boil over high heat. Cover pot and place in oven. Cook until meat is very tender when pierced with the tip of a knife, about 3 hours.

Remove oxtail pieces to a plate to cool. Discard bay leaves and thyme sprigs. Skim fat off top of stew in pot. Pick meat off bones and return to pot; discard bones and gristle. Rewarm stew over medium heat, stirring occasionally. Season with salt and pepper, if desired, and stir in parsley.