Preheat your oven to 325°F (165°C).
Season the lamb shank thoroughly on all sides with sea salt and black pepper.
Heat the olive oil in a heavy, oven-safe pot or Dutch oven over medium-high heat.
Sear the lamb shank until a deep golden-brown crust forms on all sides, then remove and set aside.
In the same pot, sauté the diced onion, carrots, celery, and minced garlic until they are fragrant and slightly softened.
Place the lamb shank back into the pot and pour in the beef broth, ensuring the liquid comes about halfway up the meat.
Add the fresh rosemary and thyme sprigs to the liquid.
Cover the pot with a tight-fitting lid and transfer to the oven to braise for 2 to 2.5 hours.
In the last 45 minutes of cooking, add the cubed parsnips to the pot, nestling them into the braising liquid.
Once the meat is fork-tender and pulling away from the bone, remove from the oven and discard the herb sprigs.
Serve the lamb shank alongside the tender vegetables, drizzled with the concentrated braising juices.