In a food processor, pulse mushrooms, onion, and garlic until finely chopped. In a large skillet, heat oil over medium heat. Add mushroom mixture and thyme, stirring often, until softened, about 4 to 6 minutes. Season with 1/4 teaspoon salt and 1/8 teaspoon pepper. Remove from heat, and mix in Parmesan, bread crumbs, and parsley. Divide into 4 equal portions.