In a small pot, heat 3 inches of oil until it reads 350°F on a thermometer. Line a large plate with several layers of paper towels.
Prepare three shallow bowls. In the first, whisk the flour with the salt, black pepper and cayenne to combine. In the second, whisk the eggs. In the third, mix the bread crumbs with the Parmesan to combine.
Working in batches, dip the eggplant pieces first into the flour mixture, then into the egg and then into the bread crumbs. Continue until all the eggplant pieces are coated.
Fry the eggplant pieces until they're golden brown, 6 to 8 minutes. Remove them from the oil and transfer to the prepared plate. Drain on the paper towels.
Garnish with parsley and coarse salt, and serve immediately with a side of warm marinara sauce for dipping.
