However, Iranian officials have seized on the fact the separatists immediately made their claim of responsibility on a Saudi-linked, Farsi-language satellite news channel based in Britain.
A video on state television's website showed confused soldiers.
An adviser to the UAE crown prince had tweeted that "attacking a military target is not a terrorist act" and "moving the battle deeper inside in Iran is a declared option". Iran has blamed its Mideast archrival, the Sunni kingdom of Saudi Arabia, for funding their activity.
The president offered condolences to the relatives and friends of those killed in the attack and wished a speedy recovery to those injured, according to the Kremlin.
Several semi-official news outlets with reporters at the scene said the attackers were dressed in military fatigues and had approached the parade from a park, behind a temporary stand from where officials had been watching the ceremony.
Among those involved are members of the Revolutionary Guard, a paramilitary force answerable only to Supreme Leader Ayatollah Ali Khamenei.
"Those who give intelligence and propaganda support to these terrorists must answer for it", he said.
But the semiofficial Tasnim news agency said at least eight IRGC members had been killed.
Armed forces spokesman Brigadier General Abolfazl Shekarchi said the dead included a young girl and a former serviceman in a wheelchair.
Women and children are among the dead, as people gathered to mark the anniversary of the start of the 1980-1988 war with Iraq.
BEHRAD GHASEMIL ISNA AP A Revolutionary Guard member carries a wounded boy after the attack by unknown militants
Iran has long-standing tensions with Israel - considering it an illegitimate occupier of Muslim land - and has accused it of attempting to undermine the government. State television immediately described the assailants as "takfiri gunmen", a term previously used to describe the Islamic State group. Officials in Tehran point to comments by Saudi Crown Prince Mohammed bin Salman, now next in line to the throne in Iran's Mideast archrival.
However, under the Trump administration, these relations have deteriorated further as the American leader pulled the United States out of a nuclear agreement that had been agreed to with Iran and other nations under his predecessor, President Barack Obama, and moved to reimpose sanctions on Tehran.
Iran's Supreme Leader, Ayatollah Khamenei, has accused "regional states that are puppets of the US", while a spokesman for the armed forces said the gunmen were trained by two Gulf Arab states with links to the USA and Israel.
Khuzestan was a major battleground of the 1980s war with Iraq and the attack on the anniversary parade in Ahvaz had significant symbolic value.
Saudi Arabia and the United Arab Emirates say that Iran poses a security threat to the Middle East and tries to dominate the region.
The Islamic State extremist group and an anti-government umbrella group, Ahvaz National Resistance, both claimed responsibility for the attack.
The attack in Ahvaz took place on the eve of President Rohani scheduled departure for NY to attend the United Nations General Assembly. But Iran has repeatedly ruled out any meeting.
- December 15, 2010: Jundallah claims a suicide attack against Shiites celebrating Ashura in Chabahar in southeastern Iran.
Why are US-Iran relations so tense?
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. Separatists attacked an Iranian military parade, killing at 29 people and wounding 70 more.