The ship returned to Portsmouth several days later but despite a huge manhunt and subsequent police investigations, his fate has remained a mystery.
His disappearance was linked to former petty officer Allan Grimson, who is serving a life sentence in prison for the murder of two young men.
Grimson was serving on board the Illustrious at the same time as Mr Parkes.
A team of nine officers will search Trafalgar Cemetery in Gibraltar this week after a member of the ship’s crew gave “credible” information, said police.
In 2003, a police review concluded Mr Parkes was likely to have been murdered and searched a number of cemeteries, but to no avail.
Detective Inspector Roger Wood said: “We have a duty to the public and to Simon’s parents to follow up on this new lead and see what comes of it.
“I would like to urge anyone who was on the HMS Illustrious who knew or knew of Simon but has not yet spoken to us, or anyone who has remembered something new since talking to us last, to get in touch as any small piece of information could really help our investigation.”
Mr Parkes’ mother, Margaret Parkes, said: “After all this time we did not think that we would get another chance to find out what happened to Simon, so we are very grateful to the police for this.
“We know there’s a chance we won’t get the answers we so desperately want but it would mean the world to us if we could finally bring Simon home.”
Grimson was jailed in 2001 for the murders of naval rating Nicholas Wright, 18, from Leicestershire, and barman Sion Jenkins from Newbury, Berkshire.
He has always denied being involved in the disappearance of Mr Parkes.