Kidnapped BBC man is alive but no clue to where he is
In a statement broadcast on the BBC, the Scottish journalist's father, Graham Johnston, called for his immediate release.
"It is no way to treat a friend of the Palestinian people," he said.
"All I can say to the men that are holding Alan, 'Please, let my son go. Now. Today'."
Mr Johnston, 44, was heading towards his apartment in Gaza City on 12 March when four masked gunmen snatched him from his car.
He had been reporting from Gaza for the past three years.
The Palestinian prime minister, Ismail Haniyeh, of Hamas, has condemned the abduction and has ordered security forces to search for the kidnappers.
Mr Johnston was the latest in a string of foreign journalists to be abducted in Gaza.
Middle East Bureaux editor Simon Wilson said the BBC did not have any information regarding a motive for the kidnapping. But people involved in the search have said that the abductors have demanded money, jobs and weapons from the Palestinian government.
Mr Wilson confirmed that the BBC has had no direct contact with the kidnappers.
"We are receiving assurances that people believe he is OK," he said.
"We are grateful, but we are disappointed that we still don't have any firm knowledge of his whereabouts seven days after he was kidnapped."