Heat oven to 425°F with a rimmed baking sheet inside to preheat.
Peel the turnips and cut them into 1-inch wedges.
In a large bowl, toss turnips with olive oil, salt, pepper, thyme, garlic, lemon zest, and red pepper flakes.
Spread the turnips on the hot sheet in a single layer, cut sides down when possible.
Roast for 15 minutes. Stir and flip, then continue roasting 8–10 minutes until tender and golden.
Sprinkle Parmesan over the hot turnips and return to the oven for 2–3 minutes to melt and lightly crisp.
Finish with parsley, taste for salt, and serve hot.