Story Highlights• Parents of BBC reporter kidnapped last month in Gaza appeal for news
• Unknown militant group claims to have killed Alan Johnston
• BBC says there is "no independent verification" of statement
• Johnston, 44, was apparently kidnapped in Gaza on March 12
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(CNN) -- The parents of BBC reporter Alan Johnston, who was kidnapped last month in Gaza, have appealed for news of his condition after an unknown militant group said it had killed him.
Graham and Margaret Johnston, who live in Scotland, on Monday released a statement saying this was a "desperately worrying time" for them.
"We make a heartfelt appeal to anyone who may have knowledge of Alan's situation and well being to contact the authorities in Gaza," they said.
The British Broadcasting Corporation said Sunday it was "deeply concerned" about the statement by the Tawad and Jihad Brigades but was treating it as a "rumor with no independent verification." (Watch video showing the militant group's written statement )
The BBC is working with Palestinian officials to investigate the claim, a spokesman for the news organization said.
"For more than a day now we have been seeking independent verification and demanding urgent clarification from the Foreign Office and the Palestinian authorities," the corporation's Director-General Mark Thompson said at a Monday afternoon vigil for BBC staff in London.
"But right now, the report is simply rumor."
Thousands of BBC journalists and other employees fell silent across Britain at 2:15 p.m. (1315 GMT) -- the exact time on March 12, five weeks ago, when he was snatched at gunpoint as he returned to his Gaza City home from his office.
In Sunday's statement, which was faxed Sunday to various news agencies, the Tawad and Jihad Brigades promised to "distribute a video showing his killing to media channels soon."
Palestinian officials said they have no evidence or information that Johnston, who was apparently abducted at gunpoint on March 12, was killed.
Palestinian Interior Minister Hani Qawasame said Sunday that, as far as he is aware, Johnston is still alive.
Palestinian chief negotiator Saeb Erakat said no demands have been made by the kidnappers.
Johnston, 44, is the only Western journalist permanently based in Gaza. On Thursday, a month after Johnston was reported missing, the BBC organized a rally in the West Bank city of Ramallah.
Speaking at the rally, Thompson issued an appeal "to all those who may have influence with the kidnappers to use their best endeavors to secure Alan's release, safely and speedily, and to ensure his return to his family and friends as quickly as possible."
Thompson told reporters he had met the day before with Mahmoud Abbas and was told by the Palestinian president that there was "credible evidence" that Johnston was "safe and well," and that Palestinians authorities were "fully engaged with Alan's case and are working to resolve it as soon as possible."
Thompson said the BBC had had no direct contact with Johnston's captors and had received no ransom demands.
Gaza militants frequently kidnap Westerners, but they are usually released unharmed.
Since 2004, 15 journalists -- including Johnston -- have been abducted by gunmen in Gaza, according to the Committee to Protect Journalists.
All the others were released unharmed, most of them within days of their kidnapping.
Johnston has been held in captivity longer than any other journalist abducted in Gaza.
Previously, the longest time in captivity had been for Steve Centanni and Olaf Wiig, Fox News network journalists who were held for two weeks in August.
According to CPJ, most of the kidnappings appear to be the work of "private individuals or groups seeking to exploit foreign hostages for political purposes or to use them as bargaining chips to secure the release of jailed relatives or to win government jobs."
Reporter Alan Johnston, shown in Gaza in an undated photo from the BBC, was apparently abducted March 12.
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