Place one rack in upper third of oven and a second in lower third; preheat oven to 425°F. Place carrots, parsnips, sweet potatoes and turnips in a large bowl and toss together with oil, paprika, thyme, salt and pepper. Spread out on two rimmed baking sheets.
Roast for 20 minutes, stirring occasionally. Rotate pans and add garlic to one pan. Continue roasting until vegetables are just tender and slightly browned, 10 to 15 minutes more.
Toss all vegetables together with parsley, if desired, and transfer to a serving dish.