Hi
This may be due to frequent hand washing, dry skin or dehydration. I would suggest that you refrain from prolonged handwashing . This may not be something serious. Wash the hands using a mild cleanser. Moisturize the hands using a mild and light vitamin E cream to keep it from drying. Drink plenty of water and avoid exposing the hands to chemicals and substances that may cause drying of skin.
This is called Aquagenic Wrinkling of the Palms. I had this exact same condition on my palms, and so did my little sister. The solution is simple!! I did some research and found out that Drysol (typically used under the arms as an anti-persperant) works to get rid of this. You can find it at any drugstore...and fyi if you have very sweaty underarms it's a miracle drug. Just apply Drysol to your palms before bed, and within a day or two the condition should clear up! I did this and told my sister to as well and we haven't had a problem since. No idea why it works but it does!