The best thing to do would be to install a Dampp-Chaser Piano Lifesaver System in the piano. Ask your tech or the dealer about this. It is less expensive in the long run than a furnace humidifier or room humidifier/de-humidifiers. Pianos benefit from stable humidity levels, and the DC system does it's job 24/7 and 365 days a year. Fluctuating temperatures will not warp any piano parts, but fluctuating levels of relative humidity can cause tuning instability, swollen bushings/ heavy action, compression ridges in the soundboard, etc.
Winter time changes in interior humidity during the cold months are largely the result of heating the air. The heated air carries less water vapor compared to the same volume of cold air, resulting in decrease in relative humidity. If there wasn't any heating, the changes in indoor relative humidity would not be as great.