Preheat the oven to 350 degrees F. Spray a 9 by 9-inch baking dish with cooking spray.
In a small pot, heat the milk with the thyme over medium-low heat until hot.
In a medium saucepan over medium heat, melt the butter with the canola oil. Add the garlic and cook 1 minute. Add the flour and cook stirring constantly until the flour is lightly golden brown, about 2 minutes. Slowly whisk in the hot milk. Bring the mixture to a simmer and cook until very thick. Stir in the Cheddar, pumpkin pie spice and salt and pepper to taste.
Coat the bottom of the prepared baking dish with a thin layer of the sauce. Place one layer of carrots slightly overlapping in the bottom of the baking dish. Top with an overlapping layer of the parsnips. Pour a third of the sauce over top. Repeat with two more layers each of the carrots and parsnips, ending with the sauce.
In a small bowl, stir together the breadcrumbs, Parmesan, parsley and salt and pepper to taste. Evenly sprinkle over the top of the gratin.
Bake until the gratin is bubbling and the topping is browned, and when pierced through the vegetables yield easily, about 1 1/2 hours. If the top begins to brown too quickly, cover with foil. Let rest for 10 minute before serving.