Parry, who has been chairman of Future since 2001, announced his intention to step down in May.
His replacement at Future, Allen, is currently chairman of Proximagen Neuroscience, Chroma Therapeutics and ProStrakan and non-executive director of Oxford Nanopore Technologies and TMO Renewables.
Allen said: "Future's strategy has positioned it well for exciting growth in digital content and I look forward to accelerating its success."
Parry is currently a non-executive chairman at Media Square, YouGov, Mobile Streams and Shakespeare’s Globe Trust.
He took over Media Square in July 2007 as executive chairman with responsibility for turning the company around after failed acquisitions led to a loss of £20m.
Under Parry’s three-year plan, Media Square reduced its debt and the annual cost of central overheads was cut from £6million to around £2m. He stepped back to become non-executive chairman after Peter Reid took up the position of chief executive last year.
Parry was previously CEO of outdoor media companies More Group and Clear Channel, and chairman of regional newspaper group Johnston Press between 2001 and 2009.
Stevie Spring, chief executive at Future, said: "On behalf of the board I would like to thank Roger Parry for more than a decade of service to Future. Personally, I have enjoyed working with him and thank him for his guidance and support.
"We are delighted to have attracted as his successor such an experienced chairman as Peter Allen. He has an excellent track record as both director and chairman of public companies. We are very lucky to have him."
Future publishes 180 magazine titles and websites, including Total Film, Classic Rock and Xbox magazine. It sells 3.2 million magazines each month and attracts more than 23 million unique visitors to its websites.
Parry was not available for comment at the time of publication.
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