Warm the vegetable oil in a large, deep pot over medium-high heat, then season the chicken all over with 2 tsp. kosher salt and arrange it skin-side down in one layer. Cook for about 5 minutes without shifting the pieces, until the skin is deeply golden, then turn and brown the other side for about 5 minutes.
Pour in the chicken stock and 2 c. water, then add the thyme and bay leaf. Bring the pot just to a bare simmer, lower the heat to medium-low, and cook for 20 to 25 minutes, stirring occasionally, until a breast reaches 165°F in its thickest part.
Move the chicken breasts to a cutting board and keep the dark meat simmering for about 40 minutes more. When the thighs are tender, transfer them to the board and let all the chicken cool for about 10 minutes before removing and discarding the skin and bones.
Shred the warm chicken into bite-sized pieces, pulling the meat across the grain rather than chopping it. The shreds should separate easily and stay moist instead of turning into long, stringy strands.
Discard the thyme and bay leaf, then add the onion, celery, and carrot to the pot over medium-low heat. Maintain the bare simmer and stir occasionally for 7 to 8 minutes, until the vegetables are tender with a little bite left at the center.
Return the shredded chicken to the broth and stir in the black pepper, then season with the remaining 2 tsp. kosher salt as needed. Taste after each adjustment so the broth becomes well seasoned without tasting sharply salty.
Ladle the soup into bowls and finish each portion with parsley and more black pepper. The parsley should stay bright green and fresh against the golden broth.