Place carrots and parsnips in a large saucepan. Add water to cover and bring to a boil. Boil until very tender, 25 to 30 minutes. Drain well and return to the pan.
Use a potato masher or ricer to finely mash the carrots and parsnips (see Tip). Add sour cream, 2 tablespoons chives, milk, butter, salt and pepper. Cook over medium-low heat, stirring, until heated through. Transfer to a serving bowl. Sprinkle with the remaining 1 tablespoon chives.
Tip: For a smoother consistency, process the vegetables in a food processor.