Heat the oven to 425°F and set out two rimmed baking sheets. Pat the prepared vegetables dry so the oil can coat them.
Whisk olive oil, mustard, salt, pepper and thyme in a large bowl. Add carrots, parsnips and turnips and toss thoroughly.
Divide between the sheets in loose single layers. Roast for 20 minutes without crowding the pieces into piles.
Turn the vegetables, swap the sheets between oven racks and roast for 15 to 20 minutes more until browned and easily pierced.
Transfer to a warm serving dish and serve promptly; avoid covering tightly, which softens the browned edges with steam.