Preheat grill to medium heat. Rub chicken with 1 teaspoon oil, 2 teaspoons salt, and 1/2 teaspoon pepper. Brush hot grill grate with oil. Place chicken, skin side down, and lemons, cut side down, on grill grate and cover. Cook 10 minutes. Transfer lemons to a plate. Continue cooking chicken 5 minutes. Flip and cook 5 minutes more.
Meanwhile, toss squashes with 2 teaspoons oil, 1 teaspoon salt, and 1/4 teaspoon pepper. Add to grill, cover, and cook, turning squashes occasionally, until squashes are tender and browned in places and a thermometer inserted into thickest part of chicken breast (do not touch bone) reaches 165 degrees, about 10 minutes.
Squeeze juice of grilled lemons into a bowl, whisk in anchovies and remaining 7 tablespoons oil, and stir in parsley. Cut chicken into pieces and serve with vegetables and parsley dressing.