Iran has suggested in recent weeks that it could take military action in the Gulf to block other countries' oil exports in retaliation for USA sanctions meant to halt its sales of crude.
State television blamed "takfiri elements", a reference to Islamist militants, for the attack in Ahvaz, the centre of Khuzestan province, which has been the site of sporadic protests by Iran's Arab minority.
Top Iranian officials have said the USA and its regional allies are responsible for the terror attack in the Iranian city of Ahvaz on Saturday.
At least eight members of Iran's Revolutionary Guard Corps (IRGC), an elite military force, died in the attack.
Neither group has provided evidence for those claims as of yet. He did not identify the attackers, though Arab separatists claimed the assault, which killed 29 people and wounded more than 60, and Iranian officials appear to have accepted the claim.
The British charge d'affaires "was told that it is not acceptable that the spokesman for the mercenary Al-Ahwazi group be allowed to claim responsiblity for this terrorist act through a London-based TV network", said the ministry's spokesman, Bahram Ghasemi.
Iran's Foreign Minister Mohammad Javad Zarif also tweeted that "Iran will respond swiftly and decisively in defence of Iranian lives".
Tensions between mainly Shi'ite Iran and mostly Sunni Saudi Arabia have surged in recent years, with the two countries supporting opposite sides in wars in Syria and Yemen and rival political parties in Iraq and Lebanon.
An unnamed spokesman for the Revolutionary Guards blamed Arab nationalists, who he said were backed by Saudi Arabia, ISNA reported.
"Terrorists began shooting from a long distance while inside the park, at the armed forces as well as civilians watching the parade", Brig.
Iranian analysts said the attack may benefit the Guards.
After addressing a similar parade in Tehran to commemorate the start of the 1980-1988 war with Iraq, Rouhani warned that "the response of the Islamic Republic of Iran to the smallest threat will be crushing".
"Iran's answer is forthcoming within the framework of law and our national interests", he added, vowing the U.S. would regret its "aggressiveness".
In a statement, the foreign office said that Pakistan is against terrorism and stands with the families of victims in this time of grief.
"It is absolutely clear to us who committed this crime. and whom they are linked to", he said.
Security forces try to control the scene in the southwestern city of Ahvaz.
Earlier on Saturday, local media reported that in a 10-minute attack, gunmen wearing military uniforms fired at Iranian forces in a military parade.
In the last decade, mass-casualty militant attacks have been incredibly rare.
In a keynote speech, he vowed to boost Iran's ballistic missile capabilities, despite Western concerns that were cited by his USA counterpart Donald Trump in May when he abandoned a landmark nuclear deal with Tehran.
Khuzestan was a major battleground of the 1980s war with Iraq and the attack on the anniversary parade in Ahvaz had significant symbolic value.