In a high-sided skillet, cover beets and potatoes with water and bring to a boil. Season with salt and cook until tender, about 7 minutes. Drain and wipe out skillet.
Heat oil in skillet over medium-high heat. Add boiled beets and potatoes and cook until potatoes begin to turn golden, about 4 minutes. Reduce heat to medium, add onion, and cook, stirring, until tender, about 4 minutes. Adjust seasoning and stir in parsley.
Make four wide wells in the hash. Crack one egg into each and season egg with salt. Cook until whites set but yolks are still runny, 5 to 6 minutes.
Start with smaller beets; they're more tender than large ones. Golden, white, and Chioggia varieties all work well in this dish.