In a medium bowl, stir together the mashed potatoes, egg, corned beef, spinach, peas and parsley. Mix with a wooden spoon to combine. Form into golf-ball-size balls, then flatten with the palm of your hand.
In a large nonstick skillet, heat the olive oil over medium-high heat. Working in batches, add the potato cakes to the pan and cook, turning once, until golden-brown and heated through, about 5 minutes. Transfer the cakes to a paper-towel-lined plate and let cool slightly.