Rinse the salt cod under cold running water to remove surface salt. Place it in a large bowl, cover it with cold water, and refrigerate for 12–24 hours, changing the water several times. Thick pieces may need longer. Before cooking, taste a small cooked flake to check the salt level.
Place the desalted cod in a saucepan and cover it with fresh water. Bring to a gentle simmer and cook for approximately 8–12 minutes, or until the fish flakes easily. Drain thoroughly, allow it to cool slightly, and remove any skin and bones. Flake the fish into small pieces.
Place the peeled and cubed potatoes in a saucepan of water. Bring to a boil and cook for 12–15 minutes, or until fork-tender. Drain very well and allow excess steam to escape. Mash until smooth, without adding too much milk or butter.
Heat a teaspoon of oil in a small skillet. Add the chopped onion and cook for 3–4 minutes until softened. Stir in the garlic and cook for another 30 seconds. Remove from the heat and allow the mixture to cool.
In a large mixing bowl, combine the flaked cod, mashed potatoes, softened onion, garlic, beaten eggs, parsley, black pepper, paprika, and lemon juice. Mix gently until everything is evenly distributed.If the mixture feels too wet to shape, add a little flour, one tablespoon at a time. Taste a small portion before adding salt, as the cod may retain some saltiness. Divide the mixture into approximately 12 equal portions. Shape each portion into a small, slightly flattened patty. If the mixture is soft, refrigerate the shaped cakes for 20–30 minutes to help them hold their form during cooking.
Heat a thin layer of oil in a large skillet over medium heat. Arrange the cakes in a single layer, leaving space between them. Cook for approximately 3–5 minutes on each side, turning carefully, until crisp and golden brown and heated through.Avoid overcrowding the pan. Transfer the cooked cakes to a plate lined with paper towels or a wire rack while you cook the remaining batches. Arrange the warm salt cod cakes on a serving platter. Garnish with fresh parsley and lemon wedges. Serve immediately with garlic mayonnaise, tartar sauce, or a fresh green salad.