Worf returns to the Enterprise after winning a bat'leth tournament, but finds a number of incongruities awaiting him. At his surprise birthday party, the cake, guests, and gifts seem to change. Then so does the console configuration at his security post, and the clues found in the investigation the crew is pursuing. When suddenly Worf finds himself first officer under a Captain Riker -- and married to Deanna Troi -- it's clear that it's not his memory that's at fault. He is hopping around between parallel dimensions, each a reality that has unfolded according to increasingly divergent histories.
What really makes "Parallels" work as an episode is that it's barely about the scientific problem of getting Worf back to his own reality. The explanation of his displacement is some brief hand-waving about how Geordi's VISOR works, and returning things to normal is an equally short and unimportant bit of technobabble. Instead, "Parallels" is pure joy for the fans, as Braga (with input from the writing staff) plays around with various what-if scenarios, springing from past episodes like Worf's paralyzing accident and the Enterprise's encounter with the Borg.
It's not just the big moves with the characters that are fun. Spot-on flourishes can be found all throughout the episode: Worf casually talking about the maimed competitors in the bat'leth tournament; Riker's glee at springing an unwanted surprise party on Worf; Troi subtly probing if Worf's request to become Alexander's "godmother" in fact contains a marriage proposal getting lost in the translation from Klingon; Captain Riker's bittersweet reaction at seeing a long-dead Picard alive again.
Given this great material to play, the actors really rise to the occasion. Michael Dorn, Marina Sirtis, and Jonathan Frakes all subsequently went on record saying how much they disliked the romantic pairing of Worf and Troi, but I think fans (and the writers, who would continue pursuing this angle) wouldn't have responded to Worf-Troi like they did if Dorn and Sirtis didn't play it so convincingly here. (In any case, the movies would ultimately marry Riker and Troi, while Deep Space Nine married Worf and Dax.)
Other departments get in on the fun of portraying alternate realities. The bridge is redecorated in a fun way (as is the ready room under Captain Riker); Data loses his signature yellow contact lenses in one reality; the alternate communicators from "Future Imperfect" make a return; a Cardassian is placed among the bridge crew; and so much more. The visual effects department bring us fantastic images of hundreds of Enterprises and half a dozen Worfs (reportedly on time and under budget!).
Wil Wheaton returns for a fun little cameo here as Wesley Crusher, in advance of his story-concluding episode later this season. Originally, it was thought that Tasha Yar would appear, but it was decided that it could make the episode seem too similar to "Yesterday's Enterprise." Unfortunately, we don't get an appearance from Alexander (whose absence is briefly explained on screen). Nor do we see parallel Troi and Worf's two children, though I personally think seeing them even for a moment or two would have dialed up the personal stakes a bit.
- I love the design of Worf's tournament trophy, which incorporates the symbol of the Klingon Empire in a clever but subtle way.
- The episode spends no time trying to convince us that Worf might be losing his mind. Brannon Braga wisely realized that he needed to put some distance between this episode and "Frame of Mind."
- Roberto Orci, co-writer of J.J. Abrams' Star Trek, cited this episode as a way of mollifying fans about the re-boot's obliteration of decades of universe continuity. Both the new timeline and the original continue to exist, he reasoned.
- The Blu-ray collection of season seven contains a short extension within the briefing room scene where parallel realities are first explained to Worf (and the audience). Everyone gets it without the extra lines, but they do have a fun personal touch where Crusher reacts to the possibility of a universe without her. (A strong contrast to a universe where she's the only person, I suppose.)