In a large bowl, combine the cold mashed potatoes, beaten egg, 1/4 cup all-purpose flour, salt, and black pepper; mix until uniform.
If the mixture is too loose to form cakes, add additional flour a tablespoon at a time until it holds together.
Heat a thin layer of oil in a large skillet over medium heat until shimmering.
Scoop about 1/4 cup of the potato mixture per cake into the hot oil and flatten each portion slightly with the back of a spatula.
Fry the cakes 3–4 minutes per side, or until golden brown and crispy, then transfer to paper towels to drain.
Sprinkle chopped fresh parsley over the potato cakes and serve warm.