Preheat the oven to 375°F (190°C).
Grease a baking dish with butter.
Layer the sliced potatoes, carrots, and parsnips in the baking dish.
In a bowl, mix the heavy cream, cheddar cheese, and half of the Parmesan cheese together.
Pour the cheese mixture evenly over the layered vegetables.
Sprinkle the remaining Parmesan cheese on top.
Cover the baking dish with aluminum foil and bake for 30 minutes.
Remove the foil and bake for an additional 15-20 minutes until bubbly and golden brown.
Let the gratin cool for 10 minutes before serving.
Garnish with fresh thyme before serving.