A statement from a spokeswoman, sent on behalf of the family, said Parker died on Friday morning.
His career included films such as Bugsy Malone, Fame, Evita and Midnight Express – and his works won a total of 19 Baftas, 10 Golden Globes and 10 Oscars.
Parker was born in Islington, London, on February 14th, 1944, and began his career in advertising as a copywriter.
He graduated to writing and directing commercials, and in 1974 moved into long form drama when he directed the BBC film, The Evacuees, written by Jack Rosenthal.
Parker wrote and directed his first feature film, Bugsy Malone, in 1975 – a musical pastiche of Hollywood gangster films of the 1930s with a cast of children.
Parker's second film, 1977’s Midnight Express, won two Oscars, six Golden Globes and four Baftas.
In 1981, he directed Pink Floyd – The Wall, the feature film adaptation of the band’s successful rock album, which became a cult classic among music fans.
In November 1995, he was made a CBE for services to the British film industry and he received his knighthood in 2002.
Parker received the Bafta Academy Fellowship Award, the body’s highest honour, in 2013.
Sean Connery, Anthony Hopkins, Laurence Olivier, Vanessa Redgrave, Christopher Lee, Martin Scorsese and Mike Leigh have been awarded the fellowship.
In 2018, Parker donated his significant private collection of scripts and working papers to the BFI National Archive.
He is survived by his wife Lisa Moran-Parker, his children Lucy, Alexander, Jake, Nathan and Henry, and seven grandchildren.
Alan Parker made so many wonderful movies. Just wonderful. A huge talent. As I’m sure you know.
RIP Alan Parker pic.twitter.com/QAGJgW9WCg— Nick Murphy (@nickmurftweets) July 31, 2020
Film director David Puttnam, who produced some of Parker's films, was among those paying tribute.
He said: “Alan was my oldest and closest friend, I was always in awe of his talent.
“My life and those of many others who loved and respected him will never be the same again.”
Director Nick Murphy described Parker as a “huge talent” in a tweet, writing: “Alan Parker made so many wonderful movies. Just wonderful. A huge talent. As I’m sure you know. RIP Alan Parker.”
