Place a rimmed baking sheet on the bottom oven rack and preheat to 425 degrees F.
Toss the celery root with 4 tablespoons olive oil, 1 tablespoon thyme, 1/2 teaspoon paprika, and salt to taste in a bowl. Pile onto a double layer of heavy-duty foil; bring the ends together and crimp closed to seal. Put the packet on another baking sheet and roast in the middle of the oven until almost tender, about 25 minutes.
Meanwhile, toss the carrots with the remaining 2 tablespoons olive oil, 1 tablespoon thyme and 1/2 teaspoon paprika in a bowl; season with salt. Spread on the preheated baking sheet and roast until tender, about 35 minutes.
After the celery root has roasted for 25 minutes, open the foil and spread on the baking sheet; roast 15 more minutes. Toss with the carrots and parsley.