Sauté Vegetables: Heat olive oil in a large pot over medium heat. Add chopped onion and garlic, sautéing for 2-3 minutes until softened. Stir in the sliced mushrooms, thyme, salt, and pepper, and cook for another 5 minutes until the mushrooms are tender and release their moisture.
Cook the Wild Rice: Add the rinsed wild rice, broth, and water to the pot. Bring the mixture to a boil, then reduce the heat to low. Cover and simmer for 45-50 minutes, or until the rice is tender and fully cooked.
Add Cream: Stir in the half-and-half (or heavy cream) and let the soup simmer for an additional 5 minutes to thicken. Adjust the seasoning with more salt and pepper, if needed.
Serve: Ladle the soup into bowls and garnish with freshly chopped parsley. Serve warm.