The BBC said on Monday that it was increasingly concerned about Johnston's safety and urged anyone with influence in the region to do all they can to ensure his safe release.
Johnston is believed to have been abducted on March 12 as he was driving from the BBC office to his home in Gaza City.
Palestinians' friend
"Holding Alan is not doing the Palestinian people any favours. Quite the opposite," the elder Johnston said.
"This is no way to treat a friend of the Palestinian people, and all I can say to the men that are holding Alan is: please let my son go, now, today," he pleaded.
He said his son had told the family that he faced possible dangers and "he did warn us that there was a possibility of being abducted - it went with the territory".
In a statement released in London, the BBC said they were certain of his abduction and detention in the Gaza Strip although no contact had been established.
"It is disappointing that after seven days there has still been no firm word either about his whereabouts or his condition," the statement said.
"We call on everyone with influence on this situation to redouble their efforts now that Alan has been missing for more than one week."
Last week Palestinian authorities were searching for Johnston, thought to be the only Western journalist still based fulltime in Gaza amid increased fighting in recent months between rivals Hamas and Fatah.
There have been a series of abductions of foreign journalists and aid workers in Gaza in the past year, but all have been released unharmed.