Preheat your oven to 325°F. Pat the oxtails dry with paper towels, then combine flour, salt, and pepper in a shallow dish. Dredge each piece of oxtail in the seasoned flour, shaking off any excess.
Heat olive oil in a large Dutch oven over medium-high heat. Working in batches so you don't crowd the pot, brown the oxtails on all sides, about 8-10 minutes total per batch. Transfer browned meat to a plate and set aside.
Reduce heat to medium and add onions to the pot. Cook for 5 minutes, scraping up all those beautiful browned bits from the bottom. Add garlic and cook for another minute until fragrant.
Stir in the tomato paste and cook for 2 minutes. Pour in the red wine, stirring well, and let it simmer for 3-4 minutes to reduce slightly.
Return the oxtails to the pot along with beef broth, bay leaves, and thyme sprigs. Bring to a simmer, then cover with the lid and transfer to the oven.
Braise for 2.5 hours, then add carrots, celery, and potatoes. Cover and continue cooking for another 45-60 minutes until the meat is fall-off-the-bone tender and vegetables are cooked through.
Remove from oven and let rest for 10 minutes. Skim off any excess fat from the surface, remove bay leaves and thyme stems, and adjust seasoning to taste.
For the best flavor, let cool completely and refrigerate overnight. The next day, scrape off the solidified fat, reheat gently on the stovetop, and serve garnished with fresh parsley. The meat will practically melt in your mouth!