Eric's friend Amanda (who works at Gotham Gazette, a place to find information about city policies) dug up this quote from Renée Paradis at the Brennan Center:

"New York is one of only nine states in the country, however, that has a party affiliation deadline that is earlier than the voter registration deadline, requiring voters who are already registered who want to switch their party enrollment to do so well before most people are thinking of the primary election or the media is covering it. Moreover, New York is one of only five states that require even those voters who are not currently affiliated with a party to meet these early deadlines. Of these five states, New York has the longest advance deadline, which is twenty-five days before the prior general election, in early October, for both the February presidential primary in presidential election years and the regular September primary in other years." --Testimony of Renée Paradis Before the New York State Elections Committee, April 24, 2009

See also Shhh, Or Independents Might Actually Get to Vote by Ms. Paradis

Amanda also points out that a candidate is not actually required to be a member of any one party to run in that party's primary. Witness NYC Mayor Bloomberg's recent efforts to secure lines on multiple primary ballots. Mayor Bloomberg is registered as "unaffiliated" but is hoping to appear on the Independence, Republican and Democratic tickets in this year's primary elections.

More articles and commentary from Gotham Gazette on NYC elections:
http://www.gothamgazette.com/article/voting/20060929/17/1987
http://www.gothamgazette.com/article/feature-commentary/20030526/202/403
http://www.gothamgazette.com/article/voting/20030122/17/196

This answer is more or less directly from Amanda, with a little editing from Jenna, your Radical Reference Librarian (who would welcome Amanda to our ranks anytime).