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.
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.