For the roasted veggies: Preheat the oven to 450 degrees F. Line a baking sheet with parchment.
Slice the carrots and parsnips into 2-inch pieces, discarding the tops. Depending on your vegetables, you may need to slice thicker ends in half or even quarters and then slice them to achieve equal pieces. Cut the onion into thick slices. Transfer the vegetables to a large bowl.
In a small bowl, mix together the olive oil, honey, thyme, salt and pepper. Pour over the vegetables and toss to coat. Dump the vegetables onto the prepared baking sheet and spread into a single layer.
Roast, shaking the baking sheet once or twice while roasting, until the vegetables are dark and golden around the edges and tender in the centers, 40 to 45 minutes.
For the dip: While the vegetables are roasting, make the dip. Place the mayonnaise, sour cream, spinach, parsley, chives, salt, basil and lemon zest in a blender or food processor and process until smooth and green. Transfer to a serving bowl and refrigerate until ready.
For the topping: To serve, transfer the vegetables to a platter and sprinkle over the Parmesan and parsley. Set out with the dip on the side.