Melt butter in a large stockpot or Dutch oven over medium heat. Add mushrooms, onion, carrots and celery. Cook, stirring occasionally, until tender, about 8-10 minutes.
Stir in garlic and thyme until fragrant, about 1 minute.
Stir in chicken, chicken stock, wild rice, bay leaf, 1 teaspoon salt and 1/2 teaspoon pepper.
Bring to a boil; reduce heat and simmer, covered, until the chicken is cooked through and the rice is tender, about 35-45 minutes. Remove chicken and let cool before dicing into bite-size pieces, discarding bones and skin.
Stir in chicken and parsley; season with salt and pepper, to taste.