Place onion skewers on grill; cook 5 minutes. Add chicken and zucchini skewers; cook, turning occasionally, 5 minutes. Add mushroom skewers; cook until chicken and vegetables are cooked through and charred in places, about 10 minutes. Dress with lemon zest and juice, oil, and parsley. Skewers can be assembled and refrigerated up to 3 hours ahead; let sit at room temperature 30 minutes before grilling.