At least 25 people died including 12 members of the Revolutionary Guards.
Nearly half of those killed were from Ahvaz, while the others came from elsewhere in Khuzestan. The attacks in Ahvaz sent women and children fleeing alongside the soldiers once marching in the parade.
Earlier in the day, Iran's intelligence minister Mahmoud Alavi told the media the security services uncovered a large network of suspects. According to Iran, a group called the "Arab Struggle Movement for the Liberation of Al-Ahwaz" carried out the attack.
Rouhani's remarks could refer to Saudi Arabia, the United Arab Emirates or Bahrain - close USA military allies that view Iran as a regional menace over its support for militant groups across the Middle East. He went on: "We will destroy them with a strong and guerrilla-style attack, inshallah (God willing)".
U.S. Secretary of State Mike Pompeo, asked by a Fox News interviewer if the United States played any role in the attack, said: "When you have a security incident at home, blaming others is an enormous mistake".
The mourners, mostly wearing black, carried pictures of the dead along with banners reading "we will stand to the end" and "no to terrorism".
A senior UAE official denied on Sunday Iranian allegations alluding to the involvement of the UAE in an attack on a military parade in southwestern Iran.
"We will punish the terrorists, one by one", he promised the crowd.
Reuters reported that Islamic State's Amaq agency posted a video that purportedly showed three men in a vehicle on their way to the parade. On Twitter, Iran's supreme leader, Ayatollah Ali Khamenei, blamed the attack on American allies in the region, calling it "a continuation of conspiracies by US-backed regimes in [the] region which have aimed at creating insecurity in our dear country".
There has been a blizzard of furious statements from top Iranian officials, including President Hassan Rouhani, accusing Iran's adversaries the USA and Gulf states of provoking the bloodshed and threatening a tough response. "We are Muslims, they are kafirs (non-believers)", the man says.
"America is acting like a bully towards the rest of the world. and thinks it can act based on brute force", said Mr Rouhani, who engineered Iran's 2015 nuclear deal that ushered in a cautious detente with Washington before tensions resurfaced with President Donald Trump's decision in May to quit the accord. Economic protests and other demonstrations have spiked across Iran, putting new pressure on Rouhani's government. Cries and wails erupted at the sight of the casket of a local hero, 54-year-old Hossein Monjazi, a disabled war veteran and Revolutionary Guard member who had lost a leg and a hand in the Iraq-Iran war.
The father of 4-year-old victim Mohammad Taha Eghdami mourns over his body during a public funeral ceremony for those killed during an attack on a military parade over the weekend.
"Just look at the crowd", said Ahvaz resident Ghaseem Farhani.