Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.
Peel the carrots and parsnips, then slice them into 1-inch thick diagonal rounds.
Toss the vegetables on the baking sheet with olive oil, sea salt, black pepper, smashed garlic cloves, and fresh thyme sprigs.
Roast the vegetables for 25 to 30 minutes, tossing halfway through, until they are tender and caramelized on the edges.
While the vegetables roast, place the duck legs skin-side down in a cold cast-iron skillet.
Turn the heat to medium and cook for 8 to 10 minutes without moving them, allowing the fat to render and the skin to become deeply golden and crispy.
Flip the duck legs and cook for an additional 2 to 3 minutes to warm through completely.
Plate the roasted root vegetables and top with the crispy duck leg, discarding the woody thyme stems before serving.