The ANC and the National Union of Metalworkers of SA (Numsa) have demanded an apology for Zapiro's latest cartoon relating to President Jacob Zuma published in the Mail & Guardian on Friday.
"As the ANC, we are taken aback by this so-called cartoon and comment by Zapiro," said the ruling ANC, which Zuma heads.
"We find it unacceptable and shocking that after the harsh experiences that South Africa, the President, and his family experienced [a] few weeks ago, that Zapiro and the Mail and Guardian will find it appropriate to continue with the insults and hurt to the President, his family, and the broader ANC constituency."
The party was referring to a now defaced painting by Brett Murray "The Spear" which depicted Zuma with his genitals exposed.
In response to the painting the ANC launched a court application to have it removed from the Goodman Gallery and arranged a protest march to register its anger.
ANC spokesman Jackson Mthembu said the party was inundated with calls about Friday's cartoon, featuring an erect penis and a limerick about Zuma.
Calls also came from journalists, leading the ANC to believe all South Africans shared its "disgust" and were "equally offended".
An apology to Zuma, his family, the ANC constituency, and to South Africa at large was required, Mthembu said.
Numsa said: "This is nothing else but a frontal attack on the personal integrity and social standing of the ANC President Jacob Zuma camouflaged as artistic expression."
"There is nothing artistic about this latest cartoon by Zapiro, but it is littered with venomous and deep personal hatred of the incumbent President Jacob Zuma," said Castro Ngobese, Numsa spokesman.
The cartoon bordered on character assassination, he continued.
"As Numsa we can come to no other conclusion but to accept that some white progressives of yesteryear have become racist and colonialist in their outlook since the advent of our new democratic dispensation.
"The likes of Zapiro and Brett Murray can no longer hide their hatred for black revolutionaries to drive our society towards a non-racial, non-sexist, democratic, and a society free of super exploitation."
The ANC Women's League (WL) earlier expressed similar views.
"[Jonathan] Shapiro has taken his attempts at satire too far," the league said in a statement.
In the cartoon, Zapiro drew an erect penis with a face and legs and a showerhead on its head looking at itself in a mirror.
Murray's signature is in a bottom corner of the "mirror" and the name of the gallery also appears in the cartoon.
Under the heading "'The Spear' to be raised at Social Cohesion Summit", he includes a limerick referring to Zuma, which the women's league felt was an insult to people who suffered under apartheid.
"The Zapiro cartoons rely on their shock value to make an impact, but calling the president of this great nation a 'dick' is unacceptable and the WL would like to know who the 'we' he is referring to in the cartoon actually is, as the majority of the population who voted for the president clearly did not think this of Zuma."