Rinse the wild rice under cold water and set it aside. Heat the olive oil in a large pot over medium heat.
Add the onion, carrots, and celery to the pot. Cook for 5 minutes, stirring often, until the vegetables soften.
Stir in the garlic, mushrooms, and thyme. Cook for another 4 minutes until the mushrooms release their liquid and begin to brown.
Add the wild rice, low-sodium vegetable broth, bay leaf, and black pepper. Bring the mixture to a boil.
Reduce the heat to low, cover the pot, and simmer for 45 minutes until the rice is tender. Remove the bay leaf.
Stir in the chopped parsley and serve the soup warm.