2. Cover the peeled chestnuts with chicken stock or water. Add the carrot, celery stalk and thyme. Salt lightly and add a grinding of pepper. Bring to a simmer. Cook until the chestnuts are no longer raw but not soft or crumbling, about 15 to 20 minutes.
3. To prepare the Brussels sprouts: Bring a saucepan of water to a boil. Salt the water generously. (This will keep the Brussels sprouts green and bring out their flavor.) Trim the stem end of the Brussels sprouts. Add them to the boiling water and blanch until still slightly crisp, about 8 to 10 minutes. Drain. Refresh by running under cold water.
4. To prepare the parsnips: Peel the parsnips, cut them in half across the middle. Cut the fatter half of the parsnip into quarters lengthwise. Cut the thin half in half lengthwise. With a small knife, remove the core from each piece. You are trying to produce 3-4" lengths of parsnip of roughly the same width.
5. To finish the dish, heat 3 tablespoons of olive oil over medium heat. Add the parsnips, season with salt and pepper. Sauté the parsnips over medium heat, tossing occasionally, until they begin to color slightly, about 8 to 10 minutes.
6. Add the Brussels sprouts and the chestnuts. They will heat through while the parsnips finish cooking. Chestnuts may also pick up some color during this process. Remove from the heat when the parsnips are tender with streaks of golden brown. Adjust the seasoning, and serve immediately sprinkled with Italian parsley.