Preheat the oven to 350 degrees F. Drizzle 1/4 cup of the heavy cream over the bottom of a 2 1/2-quart baking dish.
Peel the potatoes, celery root and parsnips, then thinly slice on a mandolin into a large bowl; toss to combine. Spread about one-third of the potato mixture in the prepared baking dish and sprinkle with 1/2 teaspoon salt, a few grinds of pepper and one-third of the remaining heavy cream.
Top with another third of the potato mixture, then sprinkle with 1 teaspoon of the thyme and about half of the remaining heavy cream. Finish with a thick layer of potato mixture and sprinkle with 1/2 teaspoon salt and the remaining 1 teaspoon thyme. Drizzle with the remaining heavy cream, then sprinkle with the manchego.
Cover the gratin with foil and bake until the vegetables are very tender, about 1 hour 20 minutes.
Preheat the broiler and carefully position a rack about 5 inches from the heat source. Uncover the baking dish and broil until the cheese browns, about 3 minutes. Let the gratin rest for 10 minute before serving.