How to Make It
Heat the oil in a large pot over medium-high heat. Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper. Cook, skin-side down, until the skin is golden brown, 7 to 8 minutes. (The chicken will not be cooked through yet.) Transfer to a plate; reserve the pot.
Add the carrots, celery, onion, ½ teaspoon salt, and ¼ teaspoon pepper to the reserved pot. Cook, stirring occasionally, until the vegetables are tender, 4 to 6 minutes. Add the broth, barley, and chicken and bring to a boil. Reduce heat and simmer, covered, until the chicken is cooked through and the barley is tender, 30 to 40 minutes.
Meanwhile, combine the parsley, lemon zest, and ¼ teaspoon each salt and pepper in a small bowl.
Transfer the chicken to a plate, discard the skin and bones, and shred the meat. Return the meat to the pot.
Serve the soup topped with the parsley mixture with the lemon wedges alongside.