IFC has a very humorous show known as Documentary Now! The show is created by Saturday Night Live Alumni Bill Hader (who starred with Amy Schumer in Trainwreck), Fred Armisen (who also has another comedy show with Portlandia), and Seth Meyers (now on Late Night). The show is a mix of deadpan humor comedy in a mockumentary format, and it has a second season already planned.
Each episode of Documentary Now begins with Helen Mirren, who sounds like she is actually introducing a serious show on PBS. By the way, Helen will be back for Season 2, and she is in on the joke.
Every Documentary Now episode is based on a real documentary, and Season 2 will take this a step further. The imdb page for Season 2, does not have a lot of episodes on it, but it has three and the last is for Season 3, Episode 7.
According to IFC, one of the episodes will be "The Bunker", which is a take on the 1993 political documentary The War Room with James Carville and George Stephanopoulos. In this case, Hader and Armisen will portray two campaign managers working on a race for the governor of Ohio. It will be a jab at politics as well as nineties culture.
Then there is "Juan Likes Rice and Chicken", which is a satire of the 2011 hit documentary Jiro Dreams of Sushi. It is shot hot in Columbia, following the young chef Arturo, played by Armisen, as well as his brother as they learn the tricks of the trade from the no-nonsense father Juan. Juan owns a highly-acclaimed restaurant that serves a dish with a lot of precision: chicken and buttered rice.
There is also "Globesmen", a parody of the 1968 Maysles Brothers documentary Salesman that followed a quartet of door-to-door Bible salesmen. In this satire, the story is about some businessmen working to sell their best globes to a community of people who find them too expensive and just want atlases.
There is also a plan for a spoof of The Kid Stays in the Picture, which talks about the Godfather producer Robert Evans. Bill Hader plans to reprise his role as the "President of Hollywood" that he portrayed at a Comedy Central Roast of James Franco. A separate article on IFC states that they might do sports documentaries.
If you are thinking that these sketches sound funny, but not too funny, you might be right. Netflix has the first season, and much of the humor is hit or miss, and sometimes it feels like it deliberately misses. It is surprising that one of their sketches "The Eye Doesn't Lie" is based on The Thin Blue Line, a documentary that actually helped clear the subject of a murder. This feels like a topic that you wouldn't satire, but they did it.
Season 2 of Documentary Now! should be out in August on IFC.