To be clear, you could newly register to vote in NY through March 25 and still be eligible to vote in today’s primary. But they defer party affiliation changes until after a general election (i.e., if you submit a party affiliation change in February, it’s not effective until a week or two after election day in November) to prevent people from switching parties just to vote in a different party’s primary.
I have mixed feelings about this system. On the one hand, it’s not particularly democratic. On the other hand, if we’re going to have political parties, I’m not sure I have a big problem with the political parties wanting their own members choosing their nominees.