The parade in Ahvaz, a city with a population of more than a million people, was one of many happening nationwide in honor of Sacred Defense Week, marking the 38th anniversary of Iran's 8-year war with Iraq.
Ali Hosein Hoseinzadeh, deputy governor in Khuzestan province, was quoted as saying the death toll was expected to rise.
No group has said it carried out the attack, which lasted for about 10 minutes.
An anti-government Arab group -- Ahvaz National Resistance -- and the Islamic State (IS) group both claimed responsibility for the killings.
"One of the southern countries in the Persian Gulf is backing them financially, providing equipment and political support", Rouhani said.
But the US has denied responsibility and says it condemns "any terrorist attack", while a senior official for the United Arab Emirates - one of the countries Mr. Rouhani was thought to be referring to - called the accusations "baseless". The group provided no evidence for the claim, Reuters reported.
Leader of the Islamic Revolution Ayatollah Seyyed Ali Khamenei said on Saturday the terrorist attack in Ahvaz was the continuation of plots hatched by United States stooges in the region to create insecurity in the country, adding that Iran's intelligence services must swiftly find the accomplices of the "criminals" behind the attack and put them on trial.
"We convey our condolences to the people and Government of Iran", it added.
At least one child was injured in the attack
"These terrorists. were trained and organised by two".
Armed forces spokesman Brigadier General Abolfazl Shekarchi said the dead included a young girl and a former serviceman in a wheelchair.
Iran will face pressure to respond to the high-profile attack on the Revolutionary Guards, the strategic force with the greatest commitment to the clerical leadership.
"Enemies should not imagine that they can gain dignity with this sinister move. I think the Iranian people have had enough and that's where all of this is coming from".
Isis wrongly suggested the Iranian president was speaking at the parade.
"Shooting began by several gunmen from behind the stand during the parade".
But attacks on regime targets inside major cities are far rarer.
State television showed images of the immediate aftermath of the attack in Ahvaz.
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 reviewing stand from where officials had been watching the ceremony.