Combine parsley, spinach, lemon juice, pine nuts, cheese, garlic, salt, and pepper in bowl of a food processor; process until smooth, about 1 minute. With processor running, add oil; process until smooth, about 1 minute. Stir together chicken, edamame, chickpeas, and cucumber in a large bowl. Add pesto; toss to combine. Place 2/3 cup arugula in each of 6 bowls; top each with 1 cup chicken salad mixture. Serve immediately.