That site also has a bookmarklet you can drag to your toolbar or bookmarks. You can click it when someone else's link puts you at the NYT registration page, and it'll take you to a fixed-up, clickable link to the story.
(found via Geek Notes2)
If that ever stops working, there's a more complicated way. You can save this page or this page to your own computer and generate a random login every time.
cross-posted from another thread