Preheat your oven to 325°F (160°C).

In a large Dutch oven, heat the olive oil over medium-high heat. Season the lamb shanks with salt and pepper. Add them to the pot and sear on all sides until browned, about 3-4 minutes per side. Remove the shanks and set aside.

In the same pot, add the diced onion, carrots, and celery. Sauté for about 5-7 minutes, or until the vegetables are softened. Add the minced garlic and cook for an additional minute.

Pour in the balsamic vinegar, scraping the bottom of the pot to release any browned bits. Stir in the broth, chopped figs, rosemary, and crushed fennel seeds.

Return the seared lamb shanks to the pot, ensuring they are submerged in the liquid. Bring the mixture to a gentle simmer, then cover the pot and transfer it to the preheated oven. Braise for 2.5 to 3 hours, or until the lamb is fork-tender.

While the lamb is braising, bring a pot of salted water to a boil. Add the chopped parsnips and cook until tender, about 15 minutes. Drain and return to the pot. Add the heavy cream and butter, and mash until smooth. Season with salt and pepper to taste.

Once the lamb is done, remove it from the oven and let it rest for 10 minutes. Skim any excess fat from the surface of the braising liquid. If desired, you can reduce the sauce over medium heat until thickened.

Plate the lamb shanks and spoon the velvety parsnip gravy over them. Drizzle with the fig-balsamic sauce from the pot. Garnish with fresh parsley.