Sprinkle the cod fillets with salt on all sides. Set the cod on a small, paper-towel-lined tray and refrigerate, uncovered, for at least 4 hours and up to 24 hours.
Preheat the oven to 350 degrees F.
Remove the cured cod from the refrigerator and pat dry. Place on a very lightly oiled baking sheet or a large nonstick, oven-proof skillet. Sprinkle each fillet with about 1 teaspoon olive oil. Transfer to the oven and cook for 8 to 10 minutes, until just warm, gently turning the fish halfway through. (You can test this by inserting a thin metal skewer or cake tester into the fish for 5 seconds. Remove it and touch it to your lip or the underside of your wrist. When it feels warm¿not hot¿the fish is done.) Remove from the oven and allow to rest at room temperature for about 5 minutes.
Place each fillet on a plate, gently pull apart the flakes and arrange evenly on the plate. Divide half of the sliced red onions and half of the capers between the plates, and sprinkle a little more of the olive oil and about 1 teaspoon red wine vinegar on each portion. Grind some black pepper over the top. Set aside to marinate for a few minutes while you prepare the vegetables that will go with the fish.
In a bowl combine the green, red and yellow peppers, the avocado, potatoes, half of the chopped parsley and the remaining vinegar and olive oil. Season with salt and pepper to taste, and gently toss together.
To finish plating the salad: Arrange the hard-boiled eggs evenly around the plates. Spoon the marinated vegetables around each plate, along with a bit of the vinegar and olive oil from the bottom of the bowl. Garnish with a few parsley leaves if you like and serve immediately.