1. Preheat oven to 400°.
2. Place potatoes and beets in a metal baking pan lined with foil. Bake at 400° for 1 hour or until tender. Remove from oven, and cool 30 minutes. Peel potatoes and beets, and cut into 1/2-inch cubes.
3. Heat a large skillet over medium-high heat. Add oil to pan; swirl to coat. Add red onion; sauté 3 minutes or until tender. Add corned beef; cook 3 minutes. Stir in ketchup, next 3 ingredients (through pepper), potato, and beets. Cover and cook 3 minutes or until thoroughly heated. Keep warm.
4. Add water to a large skillet, filling two-thirds full; bring to a boil. Reduce heat; simmer. Break 1 egg into each of 4 (6-ounce) custard cups coated with cooking spray. Place custard cups in simmering water in pan. Cover pan; cook 8 minutes. Remove custard cups from water.
5. Divide hash evenly among 4 plates. Carefully loosen eggs from custard cups. Invert 1 egg over hash on each plate. Sprinkle parsley and black pepper evenly over each serving. Serve immediately.