Rinse the fish in cold water, then soak for 48 hours in 4 changes of cold water.
In a large bowl, combine the semolina flour, the all-purpose flour, the baking powder, baking soda, salt, 2 tablespoons olive oil and enough water so that a loose batter is formed.
In a tall-sided pot or deep fryer, heat the olive oil to 375 degrees F.
Dip each fillet in the batter, allowing the excess to drip off, then dip in the bread crumbs to coat. When the oil is at temperature, gently drop the fillets in the oil and cook until golden brown and crisp, about 5 minutes.
While the fish cooks, in a medium-sized bowl, combine the parsley, onion and tomatoes and toss with 2 tablespoons extra-virgin olive oil, lemon juice and salt and pepper to taste. Remove the fish from the oil, season with salt and pepper, briefly drain on a plate lined with paper towels, and serve with 1/4 of the salad mounded atop each portion.