Heat a tablespoon of olive oil in a large heavy-bottomed pot over medium heat. Add the chopped onion and cook, stirring occasionally, until translucent and fragrant—about 5 minutes.
Add the minced garlic to the pot and cook for another minute, until it becomes fragrant and slightly golden.
Place the bone-in chicken pieces into the pot and brown them slightly, about 8 minutes, turning occasionally to develop color.
Pour in cold water to cover the chicken by an inch, then bring the mixture to a gentle boil. Skim off any foam that rises to the surface for a clear broth.
Reduce the heat to low and let the broth simmer uncovered for about 1.5 hours, until the chicken is tender and the broth has developed a rich flavor.
Remove the chicken from the pot and let it cool slightly. Shred the meat into bite-sized pieces using two forks, discarding the bones and skin.
Strain the broth through a fine mesh strainer into a clean pot to remove solids, resulting in a clear, flavorful base.
Return the shredded chicken to the strained broth. Add sliced carrots, celery, and fresh thyme, then simmer gently for another 20 minutes until vegetables are tender.
Taste the soup and season with salt and pepper as needed. Stir in chopped parsley for a fresh herbal note.
Serve the hot chicken soup in bowls, enjoying its comforting aroma and tender ingredients—perfect with crusty bread on the side.
