Set an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Grease a 9 by 13-inch glass baking dish. Set aside.
In a large skillet, melt the butter over medium-high heat. Add the butternut squash and parsnips. Season, to taste, with salt and pepper. Cook, stirring occasionally, for 5 minutes until slightly softened. Set aside to cool.
Add the bread to a large bowl.
In another bowl mix together the heavy cream, eggs, brandy, vegetable oil, brown sugar, cinnamon and thyme. Pour the cream mixture over the bread and toss gently to coat. Add the cooled vegetables and toss until the bread is coated. Spoon the mixture into the prepared pan. Bake for 40 to 45 minutes until the bread pudding is puffed and golden.