Chop the onion, celery, and carrots into ½-inch dice (1.27 cm). Slice the mushrooms, finely chop the parsley garnish, and cut the turkey into 1-inch cubes (2.54 cm).
Heat the olive oil in the Instant Pot on high sauté for 1-2 minutes. Add the onion, carrot, celery, and mushrooms and sauté for 5-10 minutes. Stir the veggies frequently. They should be softened, not browned.
Add the broth, dry wild rice, bay leaf, thyme, sage, and rosemary. Cover the Instant Pot with the lid.
To pressure cook: Close vent on lid. Pressure cook for 15 minutes using the “rice/risotto” setting under high pressure. Allow the pressure to release naturally for 10 minutes before carefully opening the lid vent.
To slow cook: Leave vent on lid open. Cook on high 3-4 hours or low 6-8 hours until the wild rice is tender. If the rice absorbs too much liquid, add up to 1 cup of extra broth before continuing.
While the soup pressure cooks or slow cooks, whisk the milk and flour together into a smooth paste. Remove the Instant Pot lid when the soup is finished pressure cooking or slow cooking.
Put the Instant Pot on the high sauté setting. Add the cooked turkey or chicken pieces, milk mixture, cream, and Parm to the InstantPot. Remove the bay leaf and discard it. Add salt and pepper if you wish (this will likely not be necessary if you used store broth).
Cook the mixture (using the high sauté setting) for 10-15 more minutes. Stir the soup frequently to keep ingredients from sticking to the bottom of the Instant Pot. The soup will thicken and become creamier.
Ladle the soup into bowls and garnish with fresh parsley. Enjoy!