Preheat oven to 375°. Place potatoes in a large saucepan; add water to cover. Bring to a boil. Reduce heat; cook, uncovered, until almost tender, 6-8 minutes. Drain., Meanwhile, in an ovenproof 12-in. skillet, cook and crumble sausage with onion over medium heat until no longer pink, 6-8 minutes. Remove from pan with a slotted spoon; wipe skillet clean., In same pan, heat oil over medium-high heat. Add potatoes; sprinkle with salt and pepper. Cook until golden brown, 10-15 minutes, turning occasionally. Stir in sausage mixture. Remove from heat., With the back of a spoon, make 8 wells in potato mixture. Break 1 egg into each well. Sprinkle with cheese., Bake until egg whites are set and yolks begin to thicken but are not hard, 12-15 minutes. Sprinkle with parsley.