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1.Heat the oil in a large pot over medium-high heat. Season the chicken with 1/2 teaspoon salt and 1/4 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.
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2.Add the carrots, celery, onion, 1/2 teaspoon salt, and 1/4 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.
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3.Meanwhile, combine the parsley, lemon zest, and 1/4 teaspoon each salt and pepper in a small bowl.
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4.Transfer the chicken to a plate, discard the skin and bones, and shred the meat. Return the meat to the pot.
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5.Serve the soup topped with the parsley mixture with the lemon wedges alongside.
