Preheat oven to 450º. Drizzle 1 Tbsp. olive oil over insides of squash and season with salt and pepper. Place cut side down on rimmed baking sheet and cover pan tightly with foil. Roast until tender, about 35 minutes.
Meanwhile, in a medium-sized skillet, heat the remaining 2 Tbsp. olive oil. Add mushrooms, onion and thyme. Season with salt and pepper. Sauté until the mushrooms are golden and the onion is translucent, about 8 minutes.
Add rice and broth and bring to a boil. Cover and reduce heat to low. Cook until liquid is absorbed, about 20 minutes.
Remove squash from oven and turn each squash over on the baking sheet. Carefully scoop out 2 to 3 tablespoons flesh from each squash half and stir into rice. Season with salt and pepper. Divide rice mixture and place inside squash halves. Sprinkle with Parmesan and place back in oven until cheese is melted.