Preheat oven to 350 degrees F (175 degrees C). Coat bottom of a 12x12-inch casserole dish with butter.
Spread mashed potatoes into the bottom of the prepared casserole dish to a depth of about 2 inches. Press ham slices into the mashed potatoes to form cup shapes. Break an egg into each ham cup. Dot mashed potatoes with butter. Sprinkle with parsley.
Bake in the preheated oven until eggs are set, 25 to 30 minutes. Scoop ham cups and mashed potatoes onto serving plates and drizzle with Hollandaise sauce.
