Blend the arugula and garlic in a food processor until finely chopped. With the machine running, gradually add 1/2 cup of oil, processing until well blended. Transfer the pesto to a large bowl. Stir in the Parmesan. Season with salt and pepper, to taste.
Prepare the grill (high heat). Using a knife, cut the back of the shrimp. Brush the shrimp with oil. Sprinkle with salt. Grill the shrimp until just cooked through, about 3 minutes per side. Transfer the shrimp to a plate.
Meanwhile, cook the pasta in a large pot of boiling salted water until tender but still firm to the bite, stirring occasionally, about 8 minutes. Drain, reserving 1/2 cup of the cooking liquid.
Toss the pasta with the pesto in the large bowl to coat, adding the reserved cooking liquid 1 tablespoon at a time to moisten. Top with the grilled shrimp. Garnish with the lemon zest and parsley.
Recipe courtesy of Giada De Laurentiis