The cartoon, in The Australian newspaper, followed a similar drawing in an Indonesian daily portraying John Howard, the Australian prime minister and Alexander Downer, the foreign minister, as copulating dingoes.
The publications come amid tensions between Jakarta and Canberra over Australia's decision to grant refugee visas to 42 asylum-seekers from Indonesia's restive Papua province.
Andi Mallarangeng, a spokesman for the Indonesian president, said the president had not seen the Australian caricature but had laughed when told of the depiction.
"It's in poor taste. Sometimes the media, both in Indonesia and other countries, resort to poor taste, which actually demonstrates the level of their quality," Mallarangeng said.
Downer responded to the latest cartoon by saying that the Australian government in no way condoned it.
"Editors have responsibility to be mindful of the consequences of what they publish, particularly when they knowingly publish material that is likely to be found offensive in some quarters," he said.
The Australian caricature cartoon, drawn by award-winning cartoonist Bill Leak, shows Yudhoyono as a dog mounting a startled-looking Papuan dog and saying "don't take this the wrong way".
The caption under the cartoon reads "no offence intended".
On Monday, the Indonesian tabloid Rakyat Merdeka ran a front-page caricature showing Howard mounting Downer and the prime minister saying: "I want Papua!! Alex! Try to make it happen."
Howard dismissed the Indonesian cartoon, although Downer described it as grotesque and "way below standards of public taste".
The Indonesian government has been stung by the decision of Australia's immigration department to issue three-year visas to the group of Papuans, including prominent separatists and their families, who arrived by boat in northern Australia in January.