Pat the chicken thighs dry and season generously with salt and pepper. Heat the olive oil in a large Dutch oven over medium-high heat. Add the chicken skin-side down and cook undisturbed until the skin is deep golden and crisp, about 6 minutes. Flip and cook 2 minutes more, then transfer to a plate. The chicken will not be cooked through yet.

Pour off all but 2 tablespoons of fat from the pot. Reduce heat to medium and add the onion, carrots, and celery. Cook, scraping up the browned bits from the bottom, until the vegetables soften and the onion turns translucent, about 6 minutes. Stir in the garlic and cook until fragrant, about 30 seconds.

Stir in the barley, thyme, and bay leaf, then pour in the chicken broth. Nestle the chicken thighs back into the pot, skin-side up, along with any accumulated juices. Bring to a boil, then reduce heat to maintain a gentle simmer. Cover partially and cook until the barley is tender and the chicken pulls away easily from the bone, about 45 minutes.

Transfer the chicken to a cutting board. When cool enough to handle, discard the skin and bones, then shred the meat into bite-sized pieces. Return the chicken to the pot, taste and adjust seasoning with salt and pepper. Ladle into bowls and top with fresh parsley.