Steps

1. Heat 1 tablespoon of the oil in a 4-quart saucepan over medium heat. Add the chicken thighs and cook, turning once, until browned on both sides. Transfer the thighs to a plate and add the celery, onions, and garlic to the pan. Cook until tender and translucent, but not brown, about 8 to 10 minutes.
2. Place the thighs on top of the vegetables, sprinkle with the salt, and add 4 cups of the water. Simmer gently until the chicken is cooked through, about 20 minutes. If necessary, add additional water to keep the thighs submerged.
3. Remove the chicken and let cool slightly. Shred the meat into bite-size pieces. Keep the pan over medium heat, add the potatoes and parsley, and simmer until the potatoes are soft. Remove larger pieces of parsley, leaving some leaves. Add the corn and the shredded chicken to the pan and simmer gently until the corn is tender, 5 to 10 minutes.
4. Add the heavy cream and simmer just until warmed through, a minute or two. Taste and adjust for salt. Ladle into bowls and season with the pepper.