Blot grated zucchini with paper towels to remove moisture. Stir zucchini, eggs, and onion in a large bowl. Mix flour, Parmesan cheese, baking powder, salt, and oregano in a separate bowl; stir mixture into zucchini until batter is just moistened.
Heat vegetable oil in a large skillet over medium-high heat. Drop rounded spoonfuls of zucchini batter into hot oil; pan fry until golden, 2 to 3 minutes per side. Drain pancakes on a paper towel-lined plate.