Cook was born at his parents' house, "Shearbridge", in Middle Warberry Road, Torquay, Devon. He was the only son, and eldest of the three children, of Alexander Edward "Alec" Cook (1906–1984), a colonial civil servant and his wife Ethel Catherine Margaret (1908–1994), daughter of solicitor Charles Mayo. His father served as political officer and later district officer in Nigeria, then as financial secretary to the colony of Gibraltar, followed by a return to Nigeria as Permanent Secretary of the Eastern Region based at Enugu. Cook's grandfather, Edward Arthur Cook (1869–1914), had also been a colonial civil servant, traffic manager for the Federated Malay States Railway in Kuala Lumpur, Malaya; stress that he suffered, in the lead-up to an interview regarding promotion, led him to commit suicide. His wife, Minnie Jane (1869–1957; daughter of Thomas Wreford, of Thelbridge and Witheridge, Devon, and of Stratford-upon-Avon, of a prominent Devonshire family traced back to 1440), kept this fact secret; Peter Cook only discovered the truth when later researching his family.