
Season the chicken pieces with salt and pepper. In a Dutch oven, heat 2 tablespoons oil over medium-high heat. Working in batches and adding more oil as needed, brown the chicken, 3 to 5 minutes per batch; set aside.
Add the remaining tablespoon oil to the drippings in the pan. Add the celery and onion and cook until beginning to soften, about 3 minutes. Stir in the garlic, ginger, and turmeric and cook until fragrant, 30 seconds to 1 minute. Sprinkle the flour over the vegetables and cook, stirring, for 1 minute. Add the chicken broth, potatoes, carrots, parsnips, 1 ½ teaspoons salt, and ½ teaspoon pepper and stir to combine.
Add the chicken back to the pot, bring to a boil, then reduce to a simmer. Cover and cook until the vegetables are tender and the chicken is cooked through, about 45 minutes.
Remove the chicken to a cutting board and let cool slightly, about 10 minutes. Shred or cut the meat into bite-size pieces, discarding the skin and bones. Stir the chicken, heavy cream, and parsley into the stew. Taste and adjust the seasoning with salt and pepper. Serve sprinkled with more parsley.