Mais oui! Tres bien, mon ami. 
An example of this would be Terry Pratchett’s Nomes trilogy.
Book 1 is a straight narrative. Book 2 has some characters who disappear in the middle of the book, and return at the end. Book 3 tells what happened to those characters in the middle.
Rosencrantz and Guildenstern are Dead would seem to be a perfect example, but I have no idea if there is a name for it.
When done so badly that it does nothing to advance the story and just puts the audience to sleep, this technique is called a ny-quel.
Rosencrantz and Guildenstern Are Dead was one of the first things that popped into my head, too, but on reflection I think it’s a clever disguise. It’s not really about the events of Hamlet, it’s about particle physics and what it must feel like to be Schrodinger’s cat.
Another example, at least in book and play, would be the Wizard of Oz and Wicked (told from the wicked witch POV).
The last 2 books of the Mutiny on the Bounty trilogy (Men Against the Sea and Pitcairn’s Island) take place at the same time, the first recounting Bligh’s voyage after the mutiny and the second the mutineers.
Piers Anthony’s Incarnations of Immortality series would be an example of parallel. Many of the events featured over the 7 book series cover the same time period.
A Horse and His Boy from the Chronicles of Narnia would be a mid-quel of sorts.
French director Lucas Belvaux did the same thing with his trilogy of films: On the Run, An Amazing Couple, and After the Life. All 3 films recount many of the same events and include the same core of 6 or 7 characters whose paths cross and intertwine, but each film is a completely different genre (thriller, romantic comedy, relationship drama) because each film focuses on a different set of characters. Not very well known in the states, but a terrific set of films.
Jane Eyre and Wide Sargasso Sea?
Susan
Two of the Saw movies (two and three, I think? I haven’t seen them, but I heard about it) take place at the same time and have some overlapping characters.
Here’s a bunch of different quels.
There is some overlap running between all four Saw movies. I would think of them more as sequels with some flashbacks/insight into prior events rather than parallels or other sequel varients.
I just checked the Wikipedia entries, and it says Saw IV runs concurrently with the events in Saw III (and the entries for each film give enough plot description that this appears to be true). All the films obviously are sequels, but Saw IV appears to specifically show “the other side” of some of the events of Saw III.
I can’t believe I’m arguing about movies that I’ve never seen, and don’t want to see.
Another example is The Horse and His Boy, from the Narnia series. Takes place during the Pevensie’s reign. The events are different, but we see three of the previous characters, in passing, from the POV of someone who never gets to know them well.
However, this is one of the reasons why ordering the Chronicles, uh…chronologically…does not work. In LWW, the story ends with them back in England, children again, and the circle is closed. It’s awkward to flash back in the very next book, and I think it’s important to have seen the Pevensies mature a bit in PC and VotDT before you see them as adults. Plus, Lewis hypes the story of Prince Cor twice in SC. Chrono ordering makes SC next-to-last, at which point you’ll be thinking, “Whaddya mean, I must read it sometime! You told it already!”
You’re right, sorry. I was thinking about parts 2 and 3 in my last statement. Bad brain moment there.
I think this is what the Japanese usually call a Gaiden - roughly “side story.”
The novel and movie Mary Reilly, starring Julia Roberts, John Malkovich, Michael Gambon and Glenn Close, retell the story of Jekyll and Hyde from the POV of his maid.
I’ll vote for sidequel. It sounds like a real word and the meaning is clear from the name.
Would Lola rennt qualify for this? The same story is told three times in one movie from the same point of view, but with a small change in the beginning of each telling that affects the outcome.
And of course, The Crimson Permanent Assurance.