Rinse the wild rice under cold water. In a pot, combine the wild rice and broth. Bring to a boil, then reduce heat, cover, and simmer for about 45-60 minutes or until the rice is tender.
In a large pot, heat olive oil over medium heat. Add chopped onion, diced carrots, and diced celery. Sauté until the vegetables are softened.
Add minced garlic and sliced mushrooms to the pot. Cook until the mushrooms release their moisture and become golden brown.
Pour in the cooked wild rice (with broth), dried thyme, bay leaf, salt, and black pepper. Bring the soup to a simmer.
Add milk and heavy cream (if using). Simmer for an additional 10-15 minutes to allow the flavors to meld.
Adjust the seasoning to taste and remove the bay leaf before serving.
Serve the soup hot, garnished with fresh parsley.