For the herbed yogurt dip: Combine the yogurt, mint, parsley, dill, garlic, lemon zest and juice, 1 teaspoon salt and 1/2 teaspoon pepper in a medium bowl and mix well; taste to check the seasonings. Cover with plastic wrap and refrigerate until ready to serve.
For the roasted veggies: Preheat the oven to 475 degrees F.
Arrange the parsnips and carrots on a baking sheet in a single layer. Arrange the zucchini and onions on a second baking sheet in a single layer. Drizzle both with the olive oil and sprinkle with salt and pepper.
Roast the parsnips and carrots until lightly browned but still firm, about 10 minutes, roast the zucchini and onions until lightly browned but not soggy, 7 to 8 minutes. Transfer to a ziptop bag and refrigerate until cold.
Serve the roasted veggies cold with the herbed yogurt dip.