(Adds vigils in Britain, Brussels paragraphs 14-15)

By Kate Kelland

LONDON, April 16 (Reuters) - The parents of BBC journalist Alan Johnston, kidnapped in Gaza more than a month ago, appealed on Monday for news of his condition after an Islamist group said it had killed him.

Prime Minister Tony Blair's office said it was urgently seeking more information on Johnston, who has not been heard from since his car was found abandoned on March 12.

Graham and Margaret Johnston, who live in Scotland, said in a statement this was a "desperately worrying time" for them.

"We make a heartfelt appeal to anyone who may have knowledge of Alan's situation and wellbeing to contact the authorities in Gaza," they said.

"Our son has lived and worked amongst the people of Gaza for the last 3 years to bring their story to the outside world -- and we ask every one of them to help end this ordeal."

An e-mail sent to media organisations on Sunday, issued in the name of the Tawhid and Jihad Brigades, said the group had killed Johnston and would release a video of the killing later.

The group was hitherto unknown in Gaza, but its name is similar to that used by movements elsewhere affiliated to al Qaeda Islamists.

The e-mail blamed the British and Palestinian authorities for failing to meet demands that prisoners be freed from Israeli jails. No such demand has been made publicly since Johnston disappeared.

Blair's official spokesman said British officials were "actively engaged" in investigating reports of Johnston's killing.

"We continue to work extremely closely with the Palestinian Authority and we are urgently seeking further information from them and a number of other sources on these reports," he said.

VIGILS

The BBC was not able to confirm the claim that Johnston, who is one of the few Western reporters based in the troubled territory, had been killed.

"The BBC has still had no independent verification of rumours concerning Alan Johnston," it said in a statement.

"We continue to be highly concerned for his safety and are working closely with the Palestinian and British authorities to seek urgent clarification."

Hundreds of BBC employees gathered outside the broadcaster's offices across Britain to show their continued support. Journalists based in Brussels and other supporters held a vigil outside the European Commission's headquarters.

"We want Alan Johnston immediately released," Margot Wallstrom, EU Communication commissioner, said in an address.

Last Thursday, a month after Johnston's disappearance, the BBC said Palestinian President Mahmoud Abbas had told it he had evidence Johnston was alive. (Additional reporting by Sophie Walker)

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