These easy-to-assemble skewers are delicious with couscous and hummus for a Mediterranean-inspired dinner.
Soak ten 12-inch wooden skewers in water 30 minutes; drain. Preheat grill over medium-high heat. Meanwhile, toss chicken with 1 tablespoon oil; season with salt and pepper. Toss vegetables with remaining 2 tablespoons oil; season with salt and pepper. Divide chicken among 4 skewers. Skewer vegetables separately, using 2 skewers for each vegetable.
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.