Prepare the turkey a day before roasting: Pulse 10 tablespoons butter, 2 tablespoons chopped thyme, the paprika, 1/2 teaspoon salt and 1/4 teaspoon pepper in a food processor until combined. Remove the neck and giblets from the turkey; discard the liver and reserve the neck and the rest of the giblets. Dry the turkey inside and out with paper towels. Loosen the skin over the breast, slide spoonfuls of the spiced butter underneath and smooth to cover the meat evenly. Rub the skin with the remaining spiced butter. Season the cavity with salt and pepper and stuff with 2 cipollinis, 3 thyme sprigs and the parsley sprigs. Refrigerate, uncovered, overnight.
Simmer the neck, chestnuts, broth and 4 cups water in a medium saucepan over low heat, partially covered, until the chestnuts are plump, 50 minutes to 1 hour. Strain, discarding the neck, and reserve the chestnuts and broth in separate bowls. (This can be done a day ahead.)
Bring the turkey to room temperature 30 minutes before roasting. Place a rack in the lowest position of the oven and remove the other racks; preheat to 400 degrees. Season the turkey all over with salt and pepper and stuff the lemon halves into the cavity. Tie the legs together with twine. Line a roasting rack with a double layer of foil and place in a roasting pan. Place the turkey, breast-side down, on the foil and roast 45 minutes. Remove the pan from the oven and turn the turkey breast-side up, discarding the foil. Roast until golden brown and a thermometer inserted into the thigh registers 160 degrees, 30 to 35 more minutes.
Meanwhile, prepare the vegetables: Melt 2 tablespoons butter in a large skillet over medium heat, then add the remaining 12 cipollinis and 2 thyme sprigs and cook until golden, about 5 minutes. Add the parsnips and cook until light brown, about 3 minutes. Add 3 cups of the reserved chestnut broth and cook until the vegetables are tender, about 12 minutes. Increase the heat to medium-high and cook until glazed, about 10 more minutes. Keep warm.
Transfer the turkey to a cutting board and place the roasting pan with drippings over medium-high heat. Whisk in the flour until browned, then whisk in 2 1/2 cups of the chestnut broth. Simmer until the gravy coats a spoon, about 5 minutes.
Melt 2 tablespoons butter in another skillet, add the mushrooms and cook until browned, about 5 minutes. Add the reserved chestnuts and cook until toasted, about 5 minutes. Season with salt and pepper, then add to the glazed vegetables. Whisk in the 2 tablespoons cold butter, then stir in the vinegar, chives and chopped parsley. Serve the turkey with the vegetables and gravy.