Winter can bring lots of snow to Central Park. This does of course look beautiful and allow for some incredible pictures. Often New York has cold, crisp days with blue skies during the Winter. January and February actually see very little rain in New York.
Winter in New York is very cold. I would suggest a long wedding dress that covers sensible shoes and maybe some very warm socks. Many brides wear a smart white coat or a fur shrug to keep the cold out. Some brides just go for a fabulous dress and take their coat off just for the ceremony and photographs. Standing around in the cold is unavoidable when getting married in Central Park in the Winter, so bare this in mind when considering your wardrobe for the day. The sun sets between 4:30 and 5:30 in the Winter, and that is when the temperature starts to drop so scheduling a wedding in the middle of the day would be best to get the most warmth and light from the sun.
From November through March skating is available in the southern end of Central Park at Wollman Rink and in the northern end at Lasker Rink. It may not be advisable to skate in a traditional ankle-length wedding dress but it might be something nice to do the day before or after your wedding. Skate and locker rentals are available.