[New York State has one of the most unforgiving rules in the country when it comes to eligibility for voting in party primaries. Anyone who wants to vote in the 2013 Democratic mayoral primary — and all of the anticipated major candidates are Democrats — but is not affiliated with a party (like Mr. Bloomberg), or wants to switch parties (as Mr. Bloomberg once did), must take action almost a year before the actual vote.
The law is rooted in the notion that closed primaries should not be raided, at the last minute, by outsiders who may want to pick, say, a weaker candidate to run against their preferred choice in a general election. And it is a law that has survived numerous court challenges, including a case from the 1970s that was upheld by the United States Supreme Court.] — http://www.nytimes.com/2012/10/05/nyregion/2013-new-york-primary-voting-requires-any-party-changes-by-oct-12.html
For the 2016 Primary you would have needed to change party registration before October 9th, 2015.
http://www.elections.ny.gov/VotingDeadlines.html
April 19, 2016 Presidential Primary Deadlines
MAIL REGISTRATION (N.Y. Election Law Section 5-210(3))
Application must be postmarked no later than March 25th and received by a board of elections no later than March 30th to be eligible to vote in the Presidential Primary.