The Greens are challenging the major parties to support a parliamentary inquiry into Australia’s participation in spying with the United States’ National Security Agency (NSA), similar to investigations being held in the UK, Spain, Germany and the US.

Guardian Australia revealed last week that the Australian spy agency, the Defence Signals Directorate, worked alongside the NSA in mounting a massive surveillance operation on Indonesia during the United Nations climate change conference in Bali in 2007.

And Fairfax newspapers had earlier revealed documents published in Germany’s Der Spiegel magazine that Australian diplomatic posts across Asia were being used to intercept phone calls and data.

The stories prompted an angry reaction from Indonesia’s foreign minister Marty Natalegawa, who suggested it could strain intelligence co-operation between the two countries.

"Looking at Indonesia and Australia and the various agreements the two countries have committed themselves, one of them obviously is agreement to exchange information, exchange even intelligence information, to address issues such as people smuggling, for example to disturb people smuggling, to disturb terrorist attacks, etc," Natalegawa said.

"Now, this information has been rather effective, rather important. We need to look at that. If Australia feels that there are ways of obtaining information other than the official ones, then one wonders where we are in terms of co-operation."

A spokesperson for foreign affairs minister Julie Bishop said on Tuesday the minister was “aware” of Natalegawa’s comments.

Co-operation on people smuggling and counter-terrorism was “to the mutual advantage of both countries”, the spokesperson said, and the minister had made this point to members of Indonesia’s parliamentary foreign affairs committee at a pre-scheduled meeting in Canberra Tuesday morning.

Malaysia has also called in the US and Australian ambassadors to complain.

Now the Greens communications spokesman, senator Scott Ludlam, is challenging both the government and the Labor opposition to back a wide-ranging parliamentary inquiry into the practices, similar to those in the UK, Spain, Germany and the US.

Germany and Brazil have also presented a draft resolution to a UN general assembly committee that calls for an end to excessive electronic surveillance, data collection and other gross invasions of privacy.

"The NSA's out-of-control surveillance culture has caused massive uproar all over the world, but the two major parties in Australia have maintained tight-lipped silence since Edward Snowden broke cover and blew the whistle," Ludlam said.

“With further revelations that Australian embassies are being used as branch offices for the NSA, the regional backlash has begun in earnest. That silence is no longer possible ... while the government will no doubt try to hide behind ‘national security’, one of the most troubling things about the NSA's surveillance programme is how much of its vast scope has nothing whatsoever to do with national security," he said.

Independent senator Nick Xenophon has also called for an inquiry. But it would require the support of a major party to eventuate. The government has refused to comment on the allegations at all. Revelations about spying at the Bali conference occurred under the Rudd government’s watch.

When he was asked about the revelations, prime minister Tony Abbott said: “Well, the thing about every Australian governmental agency is that we all operate in accordance with the law.

"Every Australian governmental agency, every Australian official, at home and abroad, operates in accordance with the law and that's the assurance that I can give people at home and abroad – our people operate in accordance with law … Now, as for the precise workings of our intelligence organisations, it's been a long-standing practice not to comment on them."

Guardian Australia has sought comment from shadow attorney general Mark Dreyfus and shadow foreign affairs spokeswoman Tanya Plibersek.