In a large bowl, stir together mashed potatoes and parsley; season with salt and pepper.
In a shallow bowl, place flour; season with salt and pepper.
In another shallow bowl, whisk egg with 1 tablespoon water.
In a third shallow bowl, place 1 ½ cups breadcrumbs.
Form potato mixture into ¾-by-2-inch logs.
Roll in flour, shaking off excess, dip in egg wash, then roll in breadcrumbs to coat.
In a heavy medium saucepan, heat 1 ½ inches vegetable oil over medium-high until hot.
In batches, fry croquettes until golden brown on all sides, about 2 minutes per batch, turning as needed.
Drain on paper towels.
