Heat a 6-quart Dutch ovenon medium-high until hot. Add thighs; cook until browned, turning once, about 8 minutes. Transfer to a bowl. Repeat with breasts.
Add broth, wine, celery, carrot, and garlic to Dutch oven. Bring to a boil; scrape any browned bits from pot; return chicken to pot, reduce heat, and simmer, skimming as necessary, for 1 hour. Transfer chicken to a plate; let cool. Transfer vegetables to another plate; reserve.
Add leeks, prunes, and barley to broth. Bring to a boil, reduce heat, and simmer until thick, about 40 minutes more. Once chicken has cooled, shred meat. Finely dice carrot and celery. Stir chicken, carrot, celery, and parsley into soup, heat through, and serve.