A "group of assailants" attacked the parade in the city of Ahvaz, held to mark the anniversary of the start of the devastating eight-year war with Saddam Hussein's Iraq in 1980, Iranian state television reported.
State television showed images of the immediate aftermath of the attack in Ahvaz.
Twenty people have been wounded in an attack on a military parade in Iran.
Hosseinzadeh also said in the clash, the security forces managed to kill two terrorists and arrest the two remaining militants.
Foreign Minister Mohammed Javad Zarif said the assault was the handiwork of "regional terror sponsors", language that usually refers to Iran's enemies Saudi Arabia and Israel, and "their US masters".
At first, those present appear to think that the gunfire could be part of the ceremony, but then quickly realize it is a hostile attack and immediately begin to scramble from the scene. He said children and journalists were casualties in the attack.
Semi-official news agency Mehr said further shooting broke out as some of the attackers who managed to escape were being chased.
Ahvaz is the capital of Iran's oil-rich Khuzestan province. Iran has been deeply involved in the fight against IS in Iraq and has aided embattled Syrian President Bashar Assad in his country's long war. However, those groups in the past previously have only attacked unguarded oil pipelines at night. At least 18 people were killed and more than 50 wounded in the attacks.
Guard spokesman Gen. Ramazan Sharif told ISNA that an Arab separatist group carried out the attack, without elaborating. The assault shocked Tehran, which largely has avoided militant attacks in the decades after the tumult surrounding the Islamic Revolution.