You're right, although I was assuming a modern day Iran that didn't suffer the Mongol demographic decline and is has larger borders than our Iran.
The problem with a modern day Iran that did not suffer from the Mongol demographic decline was that such a modern day Iran's neighbours would not suffer from a demographic decline either. And it's more difficult to accumulate large populations in the pre-Modern Middle East than people realize.
Take Galilee, for example. James A. Michener of all people explored the question of the maximum population of the Galilee in his novel "The Source". His conclusion? Galilee topped off during Byzantine times at about half a million people--all of Palestine at maybe a million and a half. And this during a time of maximum prosperity.
Egypt seems to have been limited to about 8 million people when it depended on the floods of the Nile and irrigation works in places such as El Faiyyum. Nobody ever seems to have been ambitious enough to figure out how to build polders and windmills near the Mediteranean Sea or how to build a canal from the Daimetta Mouth along the Sinai Peninsula to maximize arable land for Egypt in Pharonic or Ptolemaic or Islamic times.
And Mesopotamia?
Mesopotamia was progressively done in by soil salinification in it's formerly most fertile areas such as Sumeria. Did new areas such as the Shatt al Arab get drained and reclaimed for agriculture---maybe even rice growing? Was the Shatt ever diverted to Kuwait or Khuzistan? No it was not. Even the farms east of Bagdad salinified in the 1300s and were declining before the Mongols came in, leaving the Tigris between Mosul and Samarra and the Euphrates from Hit to Ramadi as the most fertile parts of Iraq as they seem to be today.
And Iran? Also soil salinity issues, apparently.
When people can't scratch a living from the soil and can't get jobs in the city, they do limit their birthrates. Or famine and epidemics keep their populations under control even if wars don't do it for them.
This is why even with the Colombian Exchange, Southwest Asia, including Iran were in dire ecological straits until western technology came in and allowed Iran to be self-reliant instead of having to be self-sufficient. Without a Mongol conquest to reshuffle the Mideast deck, Iran would have been hard put to expand in any direction by land except perhaps across the Persian Gulf into the Arabian Peninsula to Yemen and maybe down into East Africa.
Expansion and settlement requires a power vacuum. And in a late Medieval world sans Mongol Conquest, Iran is hemmed in the way early Modern France was IOTL. To the northwest, the Byzantine Empire still exists and is still a power. Due west is the Abbasid Caliphate, which is still intact in Iraq. Due north in the Caucasus is the Empire of Georgia, which is able to defend itself and has geography on it's side, as well as Greek Trabzon, it's ally, and Byzantium, which can come to it's aid.
Expansion due north along the western shore of the Caspian Sea to the Volga might be possible, but at some point, there's everyone from Volga Bulgaria to Patzinaks and Cumans to Tatars to deal with on the steppes. And on the northeast steppes, there is Khwarazem. Then farther to the east, the Pashtun Sultanate of Ghazni. Get past Ghazni and there's Delhi, which draws on all the resources of India, which is the same if one gets past Balochistan.
Only to the due south, conquering the small sheikhdoms and emirates of Bahrain, Qatif, Abu Zabi, Dubai, Sharjah, ect. and Oman, and moving across the forbidding Nejd to take the Holy Cities of Mecca and Medina to conquer Yemen and Hadramut are there areas that are not totally daunting to conquer. The Somali desert is doable. Ethiopia may be doable once Yemen is conquered---and the Sassanids did get as far as Yemen. And there may be lands farther than that--especially if the Shah builds ships.
Yes, in this TL, ships are the key to a Greater Persia, using the natural harbours of Bushehr, Basra (once conquered), Hormuz, and the conquered harbours of Bahrain, Qatif and Dubai and Muscat amd Sohar and even Salala, Mukalla, Aden, Mokha Hodeida and Jidda and Massuwa, once resettled with Farsi speakers.