Place a baking sheet on the middle oven rack and preheat to 475 degrees F. Toss the parsnips, carrots and turnips with 2 tablespoons olive oil, 1/2 teaspoon salt and a few grinds of pepper. Spread the vegetables on the hot baking sheet and roast until tender and browned around the edges, about 30 minutes.
Meanwhile, combine 1 cup arugula, the walnuts, parmesan, garlic, red pepper flakes, lemon zest and a big pinch of salt in a food processor. Pulse until finely chopped. With the machine running, slowly add 3 tablespoons olive oil and 2 tablespoons water.
Season the shrimp on both sides with salt and pepper. Heat a large nonstick skillet over medium-high heat; add the remaining 1 tablespoon olive oil. Add the shrimp (the skillet will be crowded) and cook until lightly browned around the edges, about 2 minutes per side; remove from the heat.
Transfer the roasted vegetables to a large bowl. Add the remaining 3 cups arugula, the shrimp and arugula pesto. Season with salt and toss. Divide among plates. Serve with lemon wedges.