In a large pot, heat the olive oil over medium heat. Add the chopped onion, carrots, and celery. Sauté for about 5 minutes until the vegetables are softened.
Add the sliced mushrooms and minced garlic to the pot. Cook for an additional 3-4 minutes until the mushrooms are tender.
Pour in the chicken broth and bring the mixture to a boil.
Once boiling, reduce the heat to low and stir in the shredded chicken, cooked rice, heavy cream, thyme, and parsley. Season with salt and pepper to taste.
Simmer the soup for 10-15 minutes, stirring occasionally, until heated through and slightly thickened.
Serve hot, garnished with additional parsley if desired.