Heat 2 tablespoons of the oil in a large skillet over medium heat.
Add the potatoes, onion, bell pepper, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until tender and golden brown, 12 to 15 minutes.
Stir in the turkey and parsley and cook, stirring occasionally, until heated through, about 3 minutes.
Meanwhile, heat the remaining tablespoon of oil in a large nonstick skillet over medium heat.
Working in 2 batches, crack the eggs into the pan and cook, covered, to the desired doneness, 2 to 4 minutes for slightly runny sunny-side-up eggs. Serve with the hash.