Sprinkle the carrots and zucchini with salt. Set aside for 15 minutes; squeeze dry.
Preheat the oven to 400 degrees F. Coat a 9-inch-square baking dish or a 9-inch cast-iron skillet with cooking spray. Whisk the cornmeal, flour, sugar, baking powder, baking soda and 1/4 teaspoon salt in a large bowl. In a separate bowl, whisk the egg, milk, sour cream and melted butter; whisk into the dry ingredients until combined. Fold in the carrots, zucchini and parsley.
Pour the batter into the prepared dish or skillet. Bake until the cornbread springs back when pressed in the center, 25 to 35 minutes. Let cool 15 minutes in the dish, then loosen the edges and slice.