Preheat oven to 425 degrees. Make the topping: Place potatoes and parsnips in a medium saucepan, cover with water by 2 inches, and bring to a simmer. Cook until fork tender, about 20 minutes. Drain, reserving 2 cups cooking liquid. Pass potatoes and parsnips through a ricer. Stir in yogurt and 3/4 teaspoon salt. Season with pepper.
Make the filling: Heat oil in a medium skillet over medium heat. Cook onion, celery, carrots, and garlic, stirring, until tender, 10 to 12 minutes.
Add turkey; cook, breaking up any large pieces, for 5 minutes. Stir in 1 1/2 teaspoons salt, the thyme, and chili powder; season with pepper. Whisk together cornstarch and reserved 2 cups cooking liquid; add to turkey. Boil for 1 minute. Stir in peas. Transfer to a 2-quart round or oval baking dish; top with potatoes. Bake until bubbling and top is browned, 35 minutes.