Preheat oven to 325°F (165°C). Season the lamb shanks generously with kosher salt and black pepper.
Heat the olive oil in a large Dutch oven over medium-high heat. Sear the shanks until browned on all sides, about 8-10 minutes. Remove and set aside.
In the same pot, add the carrots, onion, and celery. Cook until softened, about 5-7 minutes. Add garlic and cook for 1 minute more.
Stir in the tomato paste, cooking for 2 minutes until it deepens in color.
Pour in the red wine to deglaze the pot, scraping up any browned bits from the bottom. Simmer until reduced by half.
Return the shanks to the pot and pour in the beef broth. Bring to a simmer, then cover with a tight-fitting lid.
Transfer the Dutch oven to the preheated oven and braise for 2.5 to 3 hours, or until the meat is fork-tender.
Garnish with fresh parsley before serving.