What is the first mockumentary style show?

There are a lot of great mockumentary style comedy shows such as The Office, Parks and Recreation, Modern Family, etc. My question is who invented this style? Who was the first to make a show or a movie that way?

Early examples of mockumentaries.

The general genre of what you describe is known as a “mockumenary”. An early example of this genre is the 1933 film by Luis Buñuel titled, Las Hurdes: Tierra Sin Pan (Land Without Bread)

The movie is a documentary, parodying the exaggerated documentaries of travelers across the Sahara being filmed at the same time. One of Buñuel's points is that there are plenty of terrible subjects for a documentary right in Spain.

But besides that film, Orson Welles’ 1938 radio adaptation of H. G. Wells’ The War of the Worlds is considered another early mockumentary:

The first two thirds of the one-hour broadcast was presented as a series of simulated news bulletins, which suggested an actual alien invasion by Martians was currently in progress.

Mockumentaries in popular TV and film.

The 1964 Beatles film A Hard Day's Night is a mockumentary that supposedly follows the Beatles as they deal with fame and stardom but is clearly done in a very exaggerated style and was a big in-joke that was serious/fake and very meta on many levels. For example:

When Ringo is asked if he’s a mod or a rocker, he replies “Uh, no, I’m a mocker.”

The list is fairly deep but as far as TV goes there was The Rutles who were a parody band formed by some Monty Python members that mocked The Beatles and appeared on the 1975/76 comedy British comedy series Rutland Weekend Television. In 1978 a Rutles television film titled All You Need is Cash was released.

And in 1979 comedian/film-maker Albert Brooks released his first film Real Life which is a parody of the 1970s PBS series An American Family which is arguably the first reality television show ever made:

The first feature directed by Albert Brooks, who also co-authored the screenplay, it is a spoof of the 1973 reality television program An American Family and portrays a documentary filmmaker named Albert Brooks who attempts to live with and film a dysfunctional family for one full year.

Later on an episode of the 1983 British comedy series, The Comic Strip Presents… focused on a fake heavy metal band named Bad News and followed this fake band on tour. This preceded the release of the 1984 American film This is Spinal Tap which many people consider one of the most successful modern mockumentaries that inspired many others. But again, that was a film and not a TV show.

Modern TV mockumentary lineage.

The 1995 British radio mockumentary, People Like Us featured a bumbling interviewer who—in each episode—would follow and profile the lives of normal everyday people; aka: literally people like us. The radio show was made into a TV series in 1999 as well which was followed 2 years later by the British version of The Office.

So—in my humble opinion—the general concept of a mockumentary has existed for years in various forms/media. But as far as television shows specifically go we have to thank the British for creating The Rutles in 1975 which then inspired others—such as People Like Us—and lead to The Office in 2001 which then lead to Parks and Recreation and Modern Family in 2009.