Cut the salt cod into large chunks and place in a glass bowl of water, then cover with plastic wrap. Refrigerate and soak for 24-48 hours. (The longer it soaks, the less salty it will be.)
Once it's done soaking, shred the rehydrated cod into small pieces with your hands.
Bring a pot of water to a boil, add the cod, lower the heat, and simmer until it's cooked through.
Heat olive oil in a large saucepan over medium heat. Chop the onion and garlic finely and add to the pan and cook until golden. Add the cod and cook until it absorbs the oil.
Turn the heat down to low, then add the potato sticks and stir to combine. Beat the eggs lightly in a bowl, then add the eggs to the pan gradually, stirring as you go.
Continue stirring as the egg cooks, and remove the pan from the heat before the egg solidifies completely. It should be slightly runny but not totally raw.
Add salt and pepper to taste. The cod and potato sticks already have a lot of salt, so you won't need to add a lot.
Transfer to a clay or ceramic dish, garnish with olives and parsley, and serve right away.