Place the quartered potatoes in a large pot and cover with water. Add a pinch of salt. Bring to a boil over medium-high heat and cook until fork-tender, about 15-20 minutes. Drain the potatoes well.

Return the drained potatoes to the pot or a bowl. Add the butter, salt, and pepper, then mash until smooth. Allow the mashed potatoes to cool to room temperature.

Mix the cooled mashed potatoes with egg yolks, Parmesan cheese, and chopped parsley until just combined. Using your hands, form the mixture into small logs or oval shapes. Prepare three shallow dishes: one with the flour, one with the remaining beaten egg (egg wash), and one with the breadcrumbs.

Lightly coat each croquette first in flour, then dip in the egg wash, and finally roll in the breadcrumbs. Arrange the croquettes on a baking sheet lined with parchment paper.

Add enough frying oil into a medium-sized skillet to make it 1/2 inch deep. Heat the oil to 350°F, then fry the croquettes in batches until they are golden brown. Turn them occasionally while frying. Garnish with additional parsley.