Heat a large Dutch oven over medium high heat. Add 1/4 cup olive oil. While pan is heated, season oxtails generously with salt and pepper.
When the oil is hot, carefully place oxtail in pan, making sure not to overcrowd. Cook each side of the oxtail under well browned. Remove, place in bowl, and repeat with remaining oxtail.
Pour out excess oil and lightly wipe out pan with paper towel. Return the pan to medium high heat. Add remaining 1/4 cup olive oil. Add carrots, onions, celery, and garlic to pan. Stir frequently until onions begin to sweat and lightly brown.
Add tomato paste and stir into vegetables to evenly distribute. Continue to cook stirring frequently, about 3-5 minutes. Pour the wine in and bring to a boil. Let cook until reduced by half, about 10 minutes.
Pour in beef stock and bring to boil. Reduce heat to low, cover with a lid, and simmer stirring from time to time until meat is very tender and falling off the bone (about 2 hours) and sauce has reduced. Remove from heat and let cool to room temperature.
Using two forks or your fingers, remove the meat from the bones, making sure to discard any membranes, fat, or bones. Set aside in bowl.
Shred meat into bite size pieces if needed. Return meat to sauce and stir to combine. If not using the sauce the same day, cover and place in refrigerator. When ready to use, skim any fat that may have solidified on the top of the sauce, and heat over medium low heat until hot. If using same day, simply reheat heat sauce and serve over fresh parpadelle pasta and top with parmesan cheese.
