Cut the carrots and parsnips into thin strips about 2" long, then simmer in salted water until just tender. Drain and put in a serving bowl. Cover with a dish towel or foil to keep warm.
Combine the sour cream, lemon juice, mustard, brown sugar and dill in the same saucepan and heat over low heat until warm. Pour the sauce over the cooked vegetables, toss and serve.