According to local media, the terrorists also meant to attack the military commanders and officials on the podium but they were wounded by the security forces.
Yaghub Hur Totsari, spokesman for Ahvaz National Resistance, an umbrella group that claims to defend the rights of the Arab minority in Khuzestan Province, told RFE/RL's Radio Farda that the group was responsible for the attack.
The Revolutionary Guards accused Iran's arch-rival Saudi Arabia of funding the attackers, while Iran's supreme leader Ayatollah Ali Khamenei also blamed Iran's pro-US rivals.
Four militants attacked a parade commemorating the start of the 1980-1988 Iran-Iraq war in the southwestern city of Ahvaz, capital of the border province of Khuzestan.
Iran's foreign minister blamed regional countries and their "U.S. masters" for an attack on a military parade. "It is absolutely clear for us who did that, what group they are and with whom they are affiliated", Iranian President Hassan Rouhani warned before leaving for NY for the United Nations General Assembly.
The Guards have a big say in Iran's regional interests and a vast stake worth billions of dollars in Iran's economy.
Armed forces spokesman Brigadier General Abolfazl Shekarchi said the dead included a young girl and a former serviceman in a wheelchair.
"Iran will not abandon its defensive weapons. including its missiles that make America so angry", Rouhani said.
Britain said its diplomat had extended the country's condolences to Tehran and that Iranian officials were planning to lodge a formal complaint with the United Kingdom's media watchdog, Ofcom.
There has been no direct evidence linking the separatists to Saudi Arabia.
Iran's president on Sunday accused an unnamed U.S. ally of being behind a terror attack on a military parade that killed 25 people and wounded 70.
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"When you have a security incident at home, blaming others is an enormous mistake", Secretary of State Mike Pompeo told "Fox New Sunday".
That event was held on the day of the deadly attack in Ahvaz.
"The overriding concern right now is the economic crisis and the impact of USA sanctions, linked to Iran's regional contest with Riyadh and conflicts such as Syria, Yemen, and Iraq", he adds.
On Saturday, Rouhani said, "The response of the Islamic Republic of Iran to the smallest threat will be crushing".
But while violence and bomb attacks are not uncommon on both Iran's western and eastern frontiers - bordering Iraq and Afghanistan respectively - audacious and organized assaults that directly target the Islamic Republic's security services in city centers are still rare.
And Mr Rouhani said the "bully" USA, along with Gulf states it backed, had enabled the attack.
More generally, the USA is deeply suspicious of Iranian activity in the Middle East - including its influence in Syria and Yemen - and is an ally of Israel and Saudi Arabia.
The US has also expressed support for anti-government protesters in Iran.
The gunmen were trained by two Gulf Arab states and had ties to the United States and Israel, according to Shekarchi.
"America is acting like a bully toward the rest of the world.and thinks it can act based on brute force", Rouhani told reporters just before leaving Iran for the United Nations meeting in NY.
A senior United Arab Emirates official denied Iranian allegations alluding to the involvement of the UAE in training gunmen that claimed the attack.
Tensions between traditional rivals Iran and Saudi Arabia have increased in recent years, with the two countries supporting opposite sides in wars in Syria and Yemen and rival political parties in Iraq and Lebanon.