Preheat oven to 425°F. 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 ¾ teaspoon salt. Season with pepper.
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 ½ 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.
