The Cloverfield Paradox is the third installment in the franchise.
Do I need to watch the first two to make sense of the third?
As it stands, one does not need watch any of the films in any particular order, as it's The Cloverfield Paradox that gives the biggest hint as to how the Cloverfield films may relate to each other. There are times where it seems like TCP would be a "sequel" to Cloverfield, but there are some things that don't quite add up (hence the word "pradox") and many people have theories trying to find "work arounds". It may end up being a prequel to Cloverfield in some respect, as it was advertised during the Superbowl Spot as the cause or "the how" of Cloverfield, but again TCP is very vague and only lightly tangible to the first film.
With that being said, The Cloverfield Paradox has a mix (visuals, references, tones, & themes) of both films that came before it and therefor, one may not appreciate what it offers without seeing the previous two. Also there is some subject matter, that many people speculate, could tie into the upcoming fourth film allegedly titled, Overlord. And lastly, one may even get more out of it, if they are a JJ Abrams/Bad Robot fan, as it is very much like Fringe on a space station, but with a few major LOST references.
Most critics are "down" on The Cloverfield Paradox. As a longtime Bad Robot fan, I enjoyed it for what it was, but I realize that the Cloverfield films are just these little low budget sci-fi-thriller-fantasy films in the spirit of The Twilight Zone, but with a Stephen King-like story-axle. There not meant to be mind blowing, but a little remade throwback. For me, it's like watching an extended episode of an early Bad Robot TV series--a drop in the bucket, which may eventually add up to something a bit more, later depending on if these films will eventually become tighter.
UPDATE: Overlord has been confirmed to NOT be a Cloverfield film. Please disregard my comments on that specific aspect.
It isn't strictly necessary to see the other 2 films. However i would recommend watching CLoverfield as it is actually a very good film ( imo), far better than these subsequent two. These sequels are only barely related having been re-titled to fit in with the 'Cloverfield universe', a nice bit of JJ Abrahams marketing jiggery-pokery. Honestly, this last one is not good even as a standalone. you have a shot at the end which ties it to the first film... and that's it pretty much. The second film is an ok thriller with a scene near the end which gives it a belated position as a Cloverfield sequel.