The Revolutionary Guards accused Shia-dominated Iran's Sunni arch-rival Saudi Arabia of funding the attackers, while Iran's supreme leader Ayatollah Ali Khamenei also blamedIran's pro-US rivals.
The attack left at least eight members of the country's elite Revolutionary Guard dead and 20 others wounded, local media reported.
Military forces and citizens who had gathered to watch the parade were among the killed, the report said.
Both Daesh and an anti-government Arab group called the Ahvaz National Resistance, claimed responsibility for Saturday's attack.
State television aired footage of the aftermath of the assault on Ahvaz's Quds, or Jerusalem, Boulevard, which like many other places around the country saw an annual parade marking the start of Iran's long 1980s war with Iraq.
Foreign Minister Mohammad Javad Zarif said on twitter that Iran would respond "swiftly and decisively" to the attack, which he blamed on "regional terror sponsors and their USA masters".
President Hassan Rouhani vowed to deliver a "crushing response", while Iran's Supreme Leader Ayatollah Ali Khamenei linked the attack with the U.S. and its "allies in the region".
Ahvaz, the capital of Khuzestan province, is rich in oil reserves and has seen a number of attacks by the Arab separatists in the past.
Iran will face pressure to respond to the high-profile attack on the Revolutionary Guards.
The Iraqi border crossing authority said Saturday that the Iranian side was temporarily closing al-Sheeb and al-Shalamcha border crossings between the two countries after the attack.
Iran in response summoned diplomats from Denmark, the Netherlands and Britain to complain about them "hosting some members of the terrorist group" and "double standards in fighting terrorism", the foreign ministry said.
The attack undermined the Iranian government "on the day it wants to give a message to the world that it is powerful and in control", al-Tostari said.
Ahwaz is in the centre of Khuzestan province, where there have been sporadic protests by the Arab minority in mainly Shia Iran.
All four attackers were neutralized in the 10-minute gunfight. It was the deadliest such attack in Iran in almost a decade.
The State Department later offered a statement from spokeswoman Heather Nauert, who said that the US government was aware of the reports of an attack.
Echoing Zarif's comments, an Iranian army spokesman also fingered Israel and the U.S., without citing any support for his accusations.
State television blamed "takfiri elements", a reference to militants for the attack.
Who does Iran blame for the attack?
"It is absolutely clear to us who has done this, which group it is and to whom they are affiliated", Rouhani said on state television on Sunday, shortly before leaving Tehran for the United Nations General Assembly in NY.
"It is not acceptable that these groups are not listed as terrorist organizations by the European Union as long as they have not carried out a terrorist attack in Europe", Bahram Qasemi, Foreign Ministry spokesman, said, as cited by IRNA news agency.
Scott Lucas, an Iran specialist at Birmingham University in Britain and editor of the EA World View website, tells RFE/RL that Tehran's blaming of regional foes and the United States for domestic unrest is not new, while adding that it is "accentuated now by US withdrawal from the nuclear deal, imminence of comprehensive sanctions, and worsening of the Iranian economy".
Iran will be scrambling to determine the motives for the Saturday's high-profile attack as it faces growing USA pressure.
Last year, in the first deadly assault claimed by Islamic State in Tehran, 18 people were killed at the parliament and mausoleum of Ayatollah Ruhollah Khomeini, founder of the Islamic Republic.
Egypt, a key ally of Saudi Arabia, also condemned the attack.
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In videos and pictures of the aftermath circulating on social media, rescue officials can be seen shifting injured to hospitals. Meanwhile, the Takfiri Daesh terrorist group claimed responsibility for the attack but provided no evidence for the claim.
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