Preheat oven to 250°F. Place a wire rack on a rimmed baking sheet. Place eggs in a bowl. Season with salt and pepper.
Grate potatoes, parsnips, and onion with a food processor or box grater. Add garlic powder, vegetables and bread crumbs to eggs and toss to combine.
Heat 3 tablespoons oil in a large skillet over medium heat. Drop 4 (1/4-cup) spoonfuls of potato mixture into skillet; flatten to create even pancakes. Cook, turning once, until browned, 8 to 12 minutes; transfer to prepared baking sheet and keep warm in oven. Repeat with remaining potato mixture, adding more oil to skillet as needed (when getting near the end of the potato mixture, strain and discard any liquid at bottom of the bowl).
Garnish with green onion. Serve with sour cream and applesauce alongside.