Preheat the oven to 425 degrees F. Toss the carrots with 1 tablespoon olive oil, 1/2 teaspoon salt and a few grinds of pepper on a rimmed baking sheet. Tear off 3 pieces of foil; put 2 beets on each sheet. Drizzle the beets with 1 tablespoon olive oil and wrap in the foil. Transfer the carrots and beets to the oven; bake the carrots until browned and tender, 30 to 35 minutes, and bake the beets until easily pierced with a knife, 50 minutes to 1 hour.
Meanwhile, toast the pecans in a small dry skillet over low heat, stirring, about 5 minutes; let cool. Transfer to a food processor; add the parsley, mint, garlic, lemon zest, 2 tablespoons water, 1 teaspoon salt and a few grinds of pepper. Pulse a few times to make a paste. With the machine running, gradually add the remaining 1/3 cup olive oil and puree until smooth. Add the parmesan and pulse to combine.
Let the beets cool slightly, then peel and cut into wedges. Transfer to a large bowl or serving dish and add the carrots and pesto; toss to coat. Season with salt and pepper.