Preheat the oven to 450 degrees F. Toss the carrots, parsnips, red onion, garlic and 2 tablespoons olive oil on a baking sheet; season with 1/2 teaspoon salt and a few grinds of pepper. Roast, stirring halfway through, until tender and lightly browned, about 20 minutes.
Meanwhile, cook the bacon in a large nonstick skillet over medium-high heat, stirring occasionally, until crisp, 8 minutes. Transfer to a paper towel-lined plate using a slotted spoon; discard all but 1 tablespoon of the drippings from the skillet.
Whisk the remaining 2 tablespoons olive oil and the vinegar in a large bowl. Add the roasted vegetables, bacon, arugula, parsley and parmesan. Season with salt and pepper and toss.
Heat the reserved bacon drippings in the skillet over medium-high heat. Crack in the eggs and fry until just set, about 4 minutes. Serve over the salad and top with more parmesan.