Why did Mel Brooks need to get permission to film in black and white?
Because it would, in most studio executive's minds, make the movie less commercially viable.
Who did he need to get permission from?
Essentially, the studio..the company/people paying to make it.
Financing is generally arranged through studios and production companies. The executives of those companies want to make sure they are investing in a 'product' that will be commercially sucessful and will ask for changes if they feel the 'product' is threatened.
In some cases, depending on the contracts, studios have been known to take pictures away from directors (see Blade Runner). Getting permission beforehand is essential.
With The Elephant Man, Brooks defied studio convention by hiring an avant-garde director to helm a major production, shooting the film in black and white, and refusing to allow studio brass to tamper with Lynch's vision. When shown a cut of the film, Paramount executives recommended that the surreal opening and closing sequences be removed from the film. According to Cornfeld, Brooks tersely responded, "We are involved in a business venture. We screened the film for you to bring you up to date as to the status of that venture. Do not misconstrue this as our soliciting the input of raging primitives." Brooks's stubbornness was rewarded, as the film garnered eight Academy Award nominations, including Best Picture, Director and Actor.
Source - TCM.com