Position a rack in the upper third of the oven; preheat to 425 degrees F. Brush a baking sheet with olive oil; set aside. Cut the cod crosswise into 3/4-inch-thick pieces (about 16). Season generously with salt.
Heat the olive oil in a small skillet over medium heat. Add the panko and cook, tossing, until golden brown, 6 to 8 minutes. Transfer to a shallow dish and let cool completely.
Add the parmesan to the panko. Spread the flour in a separate shallow dish. Lightly beat the egg whites in a third shallow dish.
Dredge the fish in the flour, then dip in the beaten egg whites, letting the excess drip off, then dip in the panko mixture, patting to adhere. Transfer to the prepared baking sheet.
Bake until the fish is cooked through and the coating is golden brown and crisp, 15 to 18 minutes. Serve with warm marinara sauce.