Heat oil in a large saucepan over medium heat. Add mushrooms, celery, carrots and shallots and cook, stirring, until softened, about 5 minutes. Add flour, salt and pepper and cook, stirring, for 2 minutes more.

Add broth and bring to a boil, scraping up any browned bits. Add rice and reduce heat to a simmer. Cover and cook until the rice is tender, 5 to 7 minutes. Stir in turkey (or chicken), sour cream and parsley and cook until heated through, about 2 minutes more.