New York has five ballot-qualified parties which typically do not run their own nominees, but who cross-nominate a Democrat or a Republican. They are the Conservative Party (ballot-qualified since 1962); the Independence Party (ballot-qualified since 1994); the Working Families Party (since 1998); the Reform Party (since 2014); and the Women’s Equality Party (since 2014).
All five of them had at least one nominee in 2016 for Congress or legislature who was not the nominee of any other party. It is not quite fair to characterize these parties as “always” cross-nominating a Democrat or a Republican.
The Working Families and Women’s Equality Party virtually always cross-nominate a Democrat; the Reform Party virtually always cross-nominates a Republican and the Conservative Party generally cross-nominates a Republican.
The Independence Party splits its cross-nominations between the two major parties, although in 2016 it leaned Republican. Of its 144 nominees for Congress and state legislature in 2016, 92 were Republicans, 51 were Democrats, and one was solely the nominee of the Independence Party. The Independence Party has never cross-nominated a Democrat for president, but once it cross-nominated a Republican (John McCain in 2008).
In 2016, the Conservative Party’s nominees for Congress and state legislature were 128 Republicans, 6 Democrats, and 30 who were not nominees of either the Democratic or Republican Party. In 2016, the Working Families nominees for Congress and state legislature were 140 Democrats and one who was not the nominee of any other party. Also in 2016, the Women’s Equality nominees for Congress and state legislature were 76 Democrats, two Republians, and three who were not the nominee of any other party. Finally, in 2016 the Reform nominees for Congress and state legislature were 120 Republicans, three Democrats, three Conservatives, and three who were not the nominees of any other party.