Combine the zucchini, eggs, parsley, Parmigiano, and extra-virgin olive oil and stir until the zucchini is coated. Season generously with salt and pepper. Add ¾ cup of the flour just a sprinkle at a time and stir. Continue adding until the mix is the consistency of pancake batter, add more if the zucchini is very wet.
Fill a large, heavy-bottomed pan with 1/4" olive oil. Heat over medium heat until hot but not smoking. Working in batches, spoon the zucchini mixture into the pan in 2-tablespoon mounds. The fritters should be three-quarters submerged in the oil. If bits of zucchini stray, scoop them up and return them to the fritters.
Reduce the heat slightly and fry until golden brown, turning once, about 5 minutes per side. Remove from the pan with a slotted spoon and drain on paper towels. Allow the oil to return to medium heat before proceeding with the next batch.
Serve the fritters on a parchment-lined tray. The leftover fritters will keep, covered and refrigerated, up to 2 days. Reheat, wrapped in foil, in a low oven.