"These terrorists are not members of the Daesh* [terrorist group, banned in Russia] and do not belong to groups fighting the Islamic system. These people are linked to the United States and [Israeli intelligence service] Mossad. These militants were organized and trained by two countries of the Persian Gulf," Shekarchi said in an interview with a local agency.
The spokesman added that the servicemen had killed all four militants who committed the attack.
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The minister's statement came as a response to the incident reported earlier in the day when armed militants opened fire at the parade, leaving at least 25 people killed and another 60 were injured, according to media reports.
The Saudi-linked Patriotic Arab Democratic Movement in Ahwaz has reportedly claimed responsibility for the attack.
*Daesh, also known as Islamic State, ISIS or ISIL is a terrorist group, banned in the United States, Russia and many other countries.