In a large bowl, mix together the mashed potatoes, beaten egg, flour, salt, and pepper until well combined. If the mixture seems loose, add a little more flour at a time until it is stiff.
Heat oil in a large skillet over medium heat. Scoop about 1/4 cup of the potato mixture per cake into the hot oil. Flatten slightly with the back of a spatula.
Fry each side for about 3-4 minutes or until golden brown and crispy. Drain on paper towels. Sprinkle with fresh parsley when serving.