Parker was born into a working-class family in Islington, North London, the son of Elsie Ellen, a dressmaker, and William Leslie Parker, a house painter. He grew up on a council estate of Islington, which has always made it easy for him to remain "almost defiantly working-class in attitudes" said the British novelist and screenwriter Ray Connolly. Parker says that although he had his share of fun growing up, he always felt he was studying for his secondary school exams, while his friends were out having a good time. He had an "ordinary background" with no aspirations to become a film director, nor did anyone in his family have any desire to be involved in the film industry. The closest he ever came, he says, to anything related to films was learning photography, a hobby inspired by his uncles: "That early introduction to photography is something I remember."
Parker attended Dame Alice Owen's School, concentrating on science in his last year. He left school when he was eighteen to work in the advertising field, hoping that the advertising industry might be a good way to meet girls. His first job was office boy in the post room of an advertising agency. But more than anything, he says, he wanted to write, and would write essays and ads when he got home after work. His colleagues also encouraged him to write, which soon led him to a position as a copywriter in the company.