Preheat your oven to 375°F (190°C). Grease a 9x13-inch baking dish with olive oil.
In a large bowl, combine the sliced potatoes, carrots, parsnips, and onion. Season with salt, pepper, garlic powder, and thyme. Toss to coat evenly.
Layer half of the vegetable mixture in the prepared baking dish. Sprinkle half of the cheddar cheese over the vegetables.
Pour half of the heavy cream over the cheese layer. Repeat with the remaining vegetables, cheddar cheese, and heavy cream.
Top the final layer with grated Parmesan cheese.
Cover the baking dish with aluminum foil and bake for 45 minutes. Remove the foil and bake for an additional 15-20 minutes, or until the vegetables are tender and the top is golden brown.
Let the gratin cool for 5-10 minutes before serving.