4 tablespoons chopped fresh parsley, divided oil-cured black olives
Rinse the fish and place it in a bowl. Add enough cold water to cover. Chill overnight, changing the water various times.
The next day, drain the fish and remove to a large saucepan. Cover with water, bring to a boil, and simmer until the fish flakes easily, about 15 minutes. Drain and cool. Flake the fish, discarding any bones.
Heat 4 tablespoons of the oil in a heavy, large nonstick skillet over medium-high heat. Add the potatoes in sets and sauté until crisp and golden, about 7 minutes per batch. Transfer the potatoes to paper towels to drain.
Add 1 tablespoon of the oil to the same skillet. Add the onion and bay leaf and sauté until golden, about 15 minutes. Remove the bay leaf. Diminish the heat to low. Add the remaining 2 tablespoons oil to the onion slices in the skillet. Combine in the fish and potatoes. Whisk the eggs, the 1/2 teaspoon salt, and the 1/2 teaspoon pepper in a large bowl to mix. Add the egg mixture and 3 tablespoons of the parsley to the fish combination in the skillet. Cook over medium heat until the eggs are gently set, stirring irregularly, about 3 minutes. Transfer the eggs to a platter. Garnish with the olives and the remaining 1 tablespoon parsley.