Supreme Leader Ayatollah Ali Khamenei said „puppets of the US“ were trying to „create insecurity“ in Iran.
Gunmen opened fire at Revolutionary Guard troops and officials in the south-western city of Ahvaz.
Earlier an anti-government Arab group, Ahvaz National Resistance, and Islamic State (IS) both claimed the attack.
However neither group provided evidence to show they were involved.
Earlier Foreign Minister Javad Zarif blamed „terrorists paid by a foreign regime“, adding that „Iran holds regional terror sponsors and their US masters accountable“.
Iran has summoned diplomats from the UK, the Netherlands, and Denmark, accusing their countries of harbouring Iranian opposition groups, state news agency Irna reports.
„It is not acceptable that these groups are not listed as terrorist organizations by the European Union as long as they have not carried out a terrorist attack in Europe,“ said foreign ministry spokesman Bahram Qasemi.
Reports say nearly half of those killed were members of the Revolutionary Guard, who are under Mr Khamenei’s command.
Mr Khamenei did not name the „regional states“ that he believed were behind the attack.
Fars news agency said the attack started at 09:00 local time (06:30 BST), lasted about 10 minutes, and appeared to involve four gunmen.
The attackers fired at civilians and attempted to attack military officials on the podium, Fars reports.
Civilians including women and children, who were watching the military parade, were among those killed, Irna news agency said.
The victims included a four-year-old girl and a military veteran in a wheelchair, a military spokesman said.
Local journalist Behrad Ghasemi told AFP that firing continued for between 10 and 15 minutes and said at least one of the attackers was wearing a Revolutionary Guards uniform.
„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.
„Everything went haywire and soldiers started running. I saw a four-year old child get shot, and also a lady,“ he added.
All four attackers were killed, state media said.
Iran is marking the anniversary of the beginning of the 1980-88 war with Iraq with several military parades across the nation.