Iranian Foreign Minister Mohammad Javad Zarif also appeared to blame rival Riyadh for the attack.
The elite force answers to Supreme Leader Ayatollah Ali Khamenei and runs its own business empire in Iran, a major oil producer which has been relatively stable compared with Arab states that have grappled with unrest since uprisings in 2011.
Iran denies Gulf Arab accusations that it seeks to extend its sway via proxies around the Middle East, calling for states in the oil-producing region to guarantee its security without the interference of the United States and other Western powers.
Also all governmental organizations, banks, schools and universities in southeastern Khuzestan province will be closed on Monday, semi-official Tasnim news agency reported.
However, neither group provided evidence to show they were involved.
IS had claimed the attack via its propaganda mouthpiece Amaq saying it was in response to Iranian involvement in conflicts across the region.
Speaking to MEE, Rasool Hosseini, a reformist foreign policy analyst, said: "I think today's attack is in line with the United States and its regional allies' strategy to pressure Iran".
"The UAE's historical position against terrorism and violence is clear and Tehran's allegations are baseless".
An Iranian military spokesman rejected the notion that ISIS was behind the attack and said the gunmen were trained by two Gulf Arab states and had ties to the United States and Israel.
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The parade was part of nationwide ceremonies held to mark the 30th anniversary of the ending of the eight-year war with Iraq in August 1988.
Officials and an eyewitness said the gunmen were clad in Iranian military uniforms and had sprayed the crowd with gunfire using weapons they had stashed in a nearby park.
The attack was one of the worst ever against the Guards, who are the most powerful and notorious military force in the country. "It is the Americans who incite them", he said.
While he didn't name the country, the remarks are likely aimed at regional rival Saudi Arabia, which is backing President Donald Trump's drive to isolate Iran and re-impose crippling sanctions against its economy.
The Arab Struggle Movement for the Liberation of Ahwaz is classified as a terrorist group by the Iranian government.
Oman, Kuwait and Qatar issued condemnations of the attack, while Saudi Arabia, the UAE and Bahrain had yet to react on Sunday. "Instead, we will work so that the battle is for them in Iran".
Iran summoned diplomats from Britain, Denmark and the Netherlands early Sunday for allegedly harboring "members of the terrorist group" that launched the attack.
It is unacceptable that as long as these terrorist groups do not commit crimes on European soil, they will not be included in the EU's terrorist list, he highlighted.
Fars news agency quoted a local Iranian official as saying that Monday would be declared a national day of mourning.