Shred the carrots, onion, and zucchini using a grater.
Combine the vegetables, salt, and pepper in a bowl or large measuring cup.
Beat the eggs in a bowl and stir in the milk, grated Parmesan cheese, and parsley flakes.
Add the flour and beat well to make a batter.
Add the vegetables to the batter mixture and mix well.
Drop by large spoonfuls into hot oil until golden brown on both sides.
Pat dry on paper towels.
Serve warm.
