The Gulf Arab state of Qatar, which is at odds with US allies Saudi Arabia and the UAE, condemned the assault on the military parade, which wounded at least 70 people.
More generally, the United States 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, two of Iran's opponents.
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.
No evidence was provided for either of the two claims.
Ali Hosein Hoseinzadeh, deputy governor in Khuzestan province, said two attackers were killed and two were arrested, reported Al Jazeera.
Officials said the gunmen were terrorists, but so far, no specific group has come forward to claim responsibility.
Iran's Foreign Minister Mohammad Javad Zarif also tweeted that "Iran will respond swiftly and decisively in defence of Iranian lives". Iran has blamed its Mideast archrival, the Sunni kingdom of Saudi Arabia, for funding their activity.
Saudi Arabia, Bahrain and the UAE view Iran as a regional menace and have long accused it of meddling in other countries' affairs.
The assailants had hidden weapons near the parade route several days in advance, said Brigadier General Abolfazl Shekarchi, a senior spokesman for Iran's armed forces.
Videos of the attack published on social media showed the parade disintegrating into chaos as people ran from a barrage of gunshots.
The annual parade marks the start of the eight year war with Iraq in the 1980s.
Iranian Foreign Minister Mohammad Javad Zarif said "regional terror sponsors" were responsible for the attack, adding he held "their USA masters accountable".
Iran called on Denmark and the Netherlands to extradite the attack's "perpetrators and their accomplices" to stand trial, IRNA said, citing foreign ministry spokesman Bahram Qasemi.
"First we thought it's part of the parade, but after about 10 seconds we realised it was a terrorist attack as bodyguards [of officials] started shooting", he said.
Islamic State jihadists said via their propaganda mouthpiece Amaq that "Islamic State fighters attacked a gathering of Iranian forces" in Ahvaz, but the claim could not be verified and many recent IS claims have proved false.
Ahvaz is in the centre of Khuzestan province, where there have been sporadic protests by minority Arabs in predominantly Shi'ite Iran.
State television gave a toll of 29 dead and 57 wounded, while IRNA said those killed included women and children who were spectators at the parade. Rouhani said the USA withdraw from the nuclear deal was an attempt to get Iran to give up its military arsenal.
President Donald Trump decided in May to pull the United States out of the 2015 global nuclear deal with Tehran and reimpose sanctions in a bid to isolate the Islamic Republic.