In a small bowl whisk the sauce ingredients, including ¼ teaspoon salt and ¼ teaspoon pepper.
In a large nonstick skillet over medium-high heat, warm 2 tablespoons oil. Add the onion and garlic and cook until it barely starts to brown, 4 to 5 minutes, stirring frequently. Stir in the zucchini, carrots, and parsnip and cook until crisp-tender, 5 to 6 minutes, stirring frequently. Transfer to a bowl and let cool for 10 minutes. Stir in the eggs, cheese, ¼ cup flour, chives, ¾ teaspoon salt, and ¼ teaspoon pepper. If the mixture seems too loose, add another 1 to 2 tablespoons flour.
Prepare the grill for direct cooking over medium heat (about 400°F), and preheat a grill-proof griddle for about 10 minutes.
Warm 3 tablespoons oil on the griddle. Working in batches, ladle ¼ cup of the zucchini mixture for each patty onto the griddle, flattening each gently with the back of a spoon into 3-inch diameter patties. Cook over direct medium heat, with the lid closed, until the patties are browned, about 4 minutes, carefully turning once. Repeat with the remaining zucchini mixture, adding 1 to 2 tablespoons oil to the griddle for each batch. Serve the patties warm with the sauce.