Maybe with Iranian opposition forces and intellectuals fleeing to Armenia or Azerbaijan after 1979, or later after the end of the Soviet Union? The presence of Azeris in Iran could lead to a cultural crossover. Religion aside, there are also cultural similarities between Armenia and Iran.
I could see Afghanistan taking a "Iranian" turn, since parts of their population speak Dari (a variant of Farsi) and were once Persian. One might also include the Parsi in India here. By keeping the original Republic of Afghanistan going, they could develop strong ties with the Pahlavi regime of Iran. The Shah had quite a neo-Persian agenda in cultural and religious terms.
Another, more bizarre possibility: No Iranian revolution, and instead of the mujaheddin, you'd have an Iranian-influenced movement fighting against the Red Army in Afghanistan. Parts of the Soviet army - mostly Azeri, Armenian, or Tadjik - desert in large numbers and become partisans (possibly funded by the CIA and Iran), helping the Afghans and therefore creating a national pan-Iranian founding myth.