Beets add color and flavor to hash browns in this tasty one-pan breakfast. Squeeze out as much liquid as possible from the beets and the potatoes so they crisp up in the pan.
Preheat oven to 350°F. Grate potatoes, beets, and onion with the grating blade of a food processor or on the large holes of a box grater. Transfer grated vegetables to a strainer lined with a lint-free towel. Using your hands, squeeze vegetables to remove as much liquid as possible.
Cook bacon in a large ovenproof nonstick skillet over medium until crispy, about 5 minutes. Remove bacon with a slotted spoon to a plate lined with paper towels; reserve 1 teaspoon drippings in skillet. Increase heat to high; add butter and oil, and stir until warm. Add garlic to skillet, and cook, stirring often, until fragrant, about 20 seconds. Add grated vegetables; sprinkle with 3/4 teaspoon of the salt and 1/4 teaspoon of the pepper, and cook, stirring and tossing the vegetables occasionally, until vegetables start to become tender, about 5 minutes. Gently pat vegetables down into an even layer in skillet, and cook 5 minutes. Stir vegetables, and pat down again into an even layer. Cook until a crust forms on bottom, about 8 minutes. Remove from heat.
Using the back of a spoon, make 4 indentations (nests) in beet mixture in skillet. Break 1 egg into each indentation. Sprinkle eggs evenly with remaining 1/4 teaspoon each salt and pepper. Bake in preheated oven until whites are set and yolks are still runny, about 8 minutes, or to desired degree of doneness. Sprinkle top with bacon and parsley, and serve immediately.