Preheat the oven to 375 degrees F. Spray a 9-by-13-inch baking dish with nonstick cooking spray.
Combine the heavy whipping cream, chicken stock, butter, thyme, garlic and Parmesan rind in a small saucepot. Place over medium-low heat and cook just until simmering, 10 to 12 minutes. Turn off the heat and whisk in the flour. Strain the cream mixture through a colander into a container with a pour spout and discard the thyme stems, garlic and cheese rind.
Spread the potato rounds in an even layer in the bottom of the prepared baking dish. Pour a third of the cream mixture over the potatoes. Sprinkle with 1/2 teaspoon salt, 1/4 teaspoon pepper and 1/2 cup grated cheese. Next, shingle half the carrots and parsnips in alternating rows in an even layer. Pour a third of the cream mixture over the top, then sprinkle with salt, pepper and 1/2 cup cheese. Finally, shingle the remaining parsnips and carrots in alternating rows in an even layer. Use your hands to lightly press down on the vegetables to even out, if needed. Pour the remainder of the cream mixture over the top, then sprinkle with salt and pepper. Reserve the remaining cheese (about 1 cup).
Cover the baking dish tightly with aluminum foil and bake until the vegetables are fairly tender and the sauce is simmering, about 1 hour. Remove the foil and sprinkle the top of the gratin with the remaining cheese. Bake, uncovered, until the cheese is lightly browned and the vegetables are tender, another 30 minutes.
Cool for at least 10 minutes before slicing. Serve warm.