In a large heavy-bottomed stock pot or Dutch oven set over medium heat, combine the onion, carrots, celery, parsnip, thyme, butter, 1/2 teaspoon salt and 1/4 teaspoon pepper.
Stir until the butter melts then cover and cook, stirring occasionally, until the vegetables are almost tender, 10 to 15 minutes.
Add the mushrooms and cook, uncovered, stirring, until tender, 5 to 7 minutes.
Sprinkle the vegetables with the flour and cook, stirring constantly, for 2 minutes. Stir in the stock, scraping up any brown bits, and bring it to a boil, stirring. Simmer the mixture until slightly thickened, about 3 minutes.
Stir in the turkey and peas. Taste and season the mixture with salt and pepper to taste.
Transfer the filling to a 13x9" baking dish. (See Kelly's Notes.)