"The crime of killing 6-year-old children [and] the crime of shooting. women and children is nothing that".
Two of the men spoke in Arabic about jihad, while the third spoke in Farsi suggesting they were targeting the Iranian Revolutionary Guards. He went on: "We will destroy them with a strong and guerrilla-style attack, inshallah (God willing)".
An Iranian ethnic Arab opposition movement called the Ahvaz National Resistance, which seeks a separate state in oil-rich Khuzestan province, claimed responsibility for the attack.
Neither of them provided evidence.
"Hopefully we will overcome these sanctions with the least possible costs and make America regret its aggressiveness towards other countries, and particularly Iran", said Mr Rouhani.
In a rare comment on those allegations, John Bass, the U.S. ambassador in Afghanistan, told a news conference: "Let me take this opportunity, since these rumors continue to circulate, to emphatically state that the United States has not brought Daesh [an Arabic acronym for IS] to Afghanistan".
Earlier Monday, the deputy head of Iran's Revolutionary Guard Corps warned Israel and the USA that they can expect a "devastating" response from Iran, repeating accusations of their involvement in Saturday's attack. But it is unlikely the IRGC will strike any of these foes directly.
On Saturday, Trump's lawyer Rudy Giuliani promised the Iranian government's removal at an opposition conference in NY that involved the controversial dissident group Mujaheddin-e-Khalq (MEK), which has been described as a cult and accused of deadly attacks in Iran.
This muted stance was in contrast to that of other nations, which had offered their sympathies to the victims and condemnation of the attack, with several ambassadors in Tehran writing their own personal messages, while heads of state such as Russian President Vladimir Putin released official statements. "The UAE's historical position against terrorism and violence is clear and Tehran's allegations are baseless".
At Monday's funeral, however, those attending rallied around the Iranian government and its soldiers.
A senior spokesman for the Iranian Armed Forces said four terrorists were involved in the attack, noting that they had hidden their rifles in the parade venue beforehand.
Iranian President Hassan Rouhani also claimed that USA backed states in the Persian Gulf were responsible for the assault.
Tehran believes that Saudi Arabia is behind the attack, encouraged by the U.S. and Israel.
"A deadly and unforgettable revenge will be exacted.in the near future", it said.
Iran holds regular military parades to mark national anniversaries and show off its latest military hardware, notably missiles.
Kurdish rebels frequently attack military patrols on the border further north, but attacks on government targets in major cities are rare.