Soak the salted cod for 48 hours. Rinse it well under running water and place it in a bowl. Cover the cod fish with cold water and cover the bowl with plastic wrap. Refrigerate and change the water 4 times a day, until the cod is sufficiently desalted. Flake the cod into big chunks, removing any bones, and reserve.

Parboil the potatoes. Place the potatoes in a large pot and cover with salted water. Bring to a boil, reduce heat to medium-low and simmer for about 5 minutes. The potatoes should still be firm and your fork should meet resistance, as they will finish cooking in the oven. Drain and reserve.

Assemble the dish. Liberally grease a 3-qt ovenproof casserole dish with olive oil. Add 1/3 of the sliced onions, then layer half of the potatoes, half of the cod fish and another generous layer of olive oil. Repeat, finishing with a layer of onions and a good pour of olive oil. Cover with foil.

Bake in the preheated oven for 40 minutes, basting occasionally with the juices, or until the potatoes are tender and most of the excess juices have evaporated. Remove the foil, add hard-boiled eggs and olives and continue baking until golden brown, about 10 to 15 minutes more.

Remove from oven and let it cool slightly. Top with fresh parsley, if desired, and serve immediately.