The reference, to Iran's language and ancient name, was contained on white paper fliers released from Saudi aircraft in recent weeks.
As the Middle East's foremost Sunni and Shiite powers, Saudi Arabia and Iran are increasingly seen as vying for supremacy in the region, which remains beset by conflict and political turmoil.
The conflict in Yemen, where a Saudi-led coalition of Sunni Arab states launched air strikes on March 26 against Huthi Shiite rebels whom Riyadh accuses Tehran of arming, has exposed their deteriorating relations.
An AFP correspondent in Sanaa said the leaflets were dropped for two weeks during Operation Decisive Storm, the name given to the air campaign which officially ended last week. Air strikes, however, have continued.
The leaflet, in Arabic, said: "My brother of Yemen. The real goal of the coalition is to support the people of Yemen against the Persian expansion."
However, Shamkhani, a close adviser of Iran's supreme leader Ayatollah Ali Khamenei, countered by saying Iran was helping in Yemen by opposing the air strikes and providing aid.
"This is a technique that western governments used to frighten people in the cold war era," he said.
"This action by Saudi Arabia is repeating that failed western method, and shows how old the minds of Saudi politicians are."
Iran summoned the Saudi charge d'affaires in Tehran on Friday to protest after the kingdom's fighter aircraft allegedly turned back humanitarian aid flights headed for Yemen from Tehran.
"What the Saudi government is doing in Yemen resembles exactly what the Zionist regime did in Gaza. This is a massacre, a genocide," Khamenei said on April 9.
