Cook the wild rice in a saucepan according to package directions until tender, then set it aside.
Heat the olive oil in a large skillet over medium heat. Add the onion and cook until softened.
Stir in the garlic and mushrooms. Cook until the mushrooms release their liquid and begin to brown.
Add the cooked wild rice, vegetable broth, heavy cream, Parmesan cheese, thyme, salt, and pepper to the skillet. Stir well and simmer for a few minutes until the mixture thickens slightly.
Spoon the filling evenly into the prepared bell peppers and place them upright in a baking dish.
Bake at 375 degrees Fahrenheit for 25 to 30 minutes until the peppers are tender.
Remove from the oven and sprinkle with fresh parsley before serving.