Ahvaz has been the subject of several attacks by separatist groups over the past 10 years.
Khamenei ordered security forces to bring to justice the "criminals" behind one of the worst attacks ever against the Revolutionary Guards, who answer to him.
Khuzestan was a major battleground of the 1980s war with Saddam's Iraq and the attack on the anniversary parade in Ahvaz had significant symbolic value.
The Islamic State group also claimed responsibility for the attack, but provided no evidence it carried out the assault.
Iran holds regional terror sponsors and their U.S. masters accountable for such attacks. Iran sees Washington's main Arab ally in the region and a vocal critic of the nuclear deal, Saudi Arabia, as enabling US economic policies against it and trying to foment unrest in the country.
On Sunday, Iran's foreign ministry summoned a United Arab Emirates envoy over comments an unnamed official made about the attack, state TV reported, without giving further details.
Abolfazl Shekarchi, a senior spokesman for the Iranian military, said "terrorists" behind the Ahvaz attack were not members of IS or groups opposed to the Islamic establishment.
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Iranian government and military officials have pointed the finger at Gulf states, the U.S. and Israel, with all of whom Iran has longstanding tensions. State media in Saudi Arabia did not immediately acknowledge the attack, though a Saudi-linked, Farsi-language satellite channel based in the United Kingdom immediately carried an interview with an Ahvazi activist claiming Saturday's attack.
"Iran holds regional terror sponsors and their U.S. masters accountable for such attacks", he wrote on his Twitter account.
There has been a blizzard of furious statements from top Iranian officials, including President Hassan Rouhani, accusing Iran's adversaries the United States and Gulf states of provoking the bloodshed and threatening a tough response.
An adviser to the UAE crown prince had tweeted that "attacking a military target is not a terrorist act" and "moving the battle deeper inside in Iran is a declared option".
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. Rouhani said the USA withdraw from the nuclear deal was an attempt to get Iran to give up its military arsenal.
The Iraqi border crossing authority said Saturday that the Iranian side was temporarily closing al-Sheeb and al-Shalamcha border crossings between the two countries after the attack.
In 2015, under then-president Barack Obama, the United States and Iran reached a landmark nuclear deal - also signed by China, Russia, the UK, France and Germany - where Iran limited its nuclear activities in exchange for sanctions relief.
The government of Syria also expressed its "full sympathy and solidarity" with Iran, while Turkey slammed a "heinous terrorist attack".
Brigadier General Abolfazl Shekarchi was quoted by IRNA as saying the terrorists were "not from Da'esh (ISIS) or other groups fighting (Iran's) Islamic system ... but are linked to America and the [Israeli worldwide intelligence agency] Mossad".