Combine zucchini and salt in a bowl and stir to combine. Set aside for 10 minutes. Pour mixture onto a clean dish towel or cheesecloth and squeeze to drain completely.
Whisk egg in a bowl and add flour, Parmesan cheese, salt, and pepper. Stir in drained zucchini and mix well.
Heat oil in a medium-sized pan over medium-high heat. Add batter by the tablespoon. Cook fritters until golden brown on both sides, about 5 minutes per batch. Transfer to a serving plate and sprinkle with Parmesan cheese and salt. Serve immediately with sour cream.
