Place the potatoes in a large pot of water, bring the water to a boil. Let the potatoes cook until they are almost tender.
Add the fish to the pot and let the fish and potatoes cook until they are both soft. Drain well and transfer the potatoes and fish to a large mixing bowl.
Add butter, onion, parsley, and egg to the bowl; mash the mixture together. Mold the mixture into patties.
Heat oil in a large skillet over a medium-high heat. Fry the patties on both sides until golden brown. Drain on paper towels before serving.
