Season the chicken pieces with salt and pepper. Heat 2 tablespoons oil in a Dutch oven over medium-high heat. Brown the chicken, working in batches and adding more oil as needed, 3 to 5 minutes per batch; set aside.
Add the celery and onion and cook until beginning to soften, about 3 minutes. Sprinkle over the flour and stir, cooking a minute or so. Pour in the stock, season with salt and pepper and add the parsnips, carrots, turnip and rutabaga. Add the chicken back, bring to a boil and reduce to a simmer. Cover and cook until the vegetables are tender and the chicken is cooked through, about 45 minutes.
Stir in the heavy cream and parsley. Taste and adjust the seasoning with salt and pepper. Serve garnished with parsley leaves.