A journalist named Yunus Purhulu was among those killed. He died on Saturday.
He described his group as leading an umbrella organization for other Ahvaz groups, which includes the militants who carried out Saturday's attack in Ahvaz.
The group is supported by "foreign antagonists", including Saudi Arabia, the agency said.
"There are a number of nonmilitary victims, including women and children who had come to watch the parade", IRNA quoted an unnamed official source as saying. Some 60 people were wounded in the attack, majority civilians.
The British chargé d'affaires was called in along with the ambassadors of the Netherlands and Denmark after the atrocity which saw four gunmen open fire on spectators at a parade in the in the southern city of Ahvaz.
"Those who give intelligence and propaganda support to these terrorists must answer for it", he said.
U.S. Secretary of State Mike Pompeo on Sunday rejected Tehran's accusations that Washington and its allies should be held accountable for a deadly parade attack in Iran.
Iran's President Hassan Rouhani claims an unnamed US -allied country supports the group that attacked a military parade on Saturday.
With eyes on Iran, Bolton warns against Russia-Syria missile deal
The Israeli leader also agreed with Putin on continued co-ordination between the two countries' armed forces. Then, "one minute" after Israel's call, its F-16 planes struck targets in Latakia in western Syria, he said.
The IS claim initially incorrectly said the attack targeted Iranian President Hassan Rouhani. He did not identify the attackers.
Iran, Opec's third-largest oil producer, has also accused Mr Trump of orchestrating the oil price rally by imposing sanctions on Teheran and accused its regional arch-rival Saudi Arabia of bowing to United States pressure. At the same time, he said that Iran's response to "these crimes" will be "within the framework of law".
Arab Gulf nations view Iran as a regional menace and have long accused it of meddling in other countries' affairs.
Arabs are one of several major non-Persian ethnic groups in Iran that also include Kurds, Turks, and Baluch.
The country's foreign ministry also hit out at a Saudi-linked, Farsi-language satellite channel based in Britain, which aired an interview with an Ahvazi separatist claiming responsibility for the killings.
Saturday's attack in Ahvaz killed at least 25 people and wounded over 60.
The separatist group the Patriotic Arab Democratic Movement in Ahwaz claimed responsibility for this attack.