Unknown gunmen opened fire on people from behind the stand during the parade held to mark the former Iraqi dictator Saddam Hussein's invasion of Iran which led to an eight-year war in the 1980s, Press TV reported.
"There are a number of non-military victims, including women and children who had come to watch the parade", the IRNA news agency quoted an unnamed official source as saying.
Ahvaz is the capital of Iran's oil-rich Khuzestan Province. The province has a large ethnic Arab community, many of them Sunni, the Muslim denomination to which IS adheres and which is a minority in Shiite Iran.
State media say Islamist militants are thought to be behind the attack.
Meantime, media reports said that the Al-Ahvaziya terrorist group has claimed responsibility for the attack.
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A recovery operation in the Mediterranean Sea is underway, Konashenkov said. Israel has refrained from taking sides in the Syrian civil war.
State television showed images of the immediate aftermath, with paramedics seen helping someone in military fatigues lying on the ground.
Fars news agency said the attack started at 09:00 local time (06:30 BST) and appeared to involve at least two gunmen.
Iran's Foreign Minister Mohammad Javad Zarif blamed a "foreign regime" backed by the United States for paying, recruiting, and training terrorists for the assault in Ahvaz. Two of the attackers are dead, one wounded and another captured, according to its report.
The Revolutionary Guard is a paramilitary force answerable only to Supreme Leader Ayatollah Ali Khamenei. The Guard also has vast holdings in Iran's economy.
The semi-official ISNA news agency published photographs of the aftermath, with bloodied troops in dress uniforms helping each other away from the scene on Ahvaz's Quds, or Jerusalem Boulevard. However, those groups in the past previously have only attacked unguarded oil pipelines at night. The assault shocked Tehran, which has seen few militant attacks in the decades after the tumult surrounding the Islamic Revolution.