I think the reality could have been more nuanced that this.

Persians were the prominent tribe in Achaemenid empire which was destroyed by Alexander. But remainder of Persians among other tribes that formed the empire later on freed themselves from Macedonian rule and eventually and gradually pushed them out. Even though the Macedonians occupied the land, they adopted the form of government and were influenced by the culture and Persian way of life. They themselves also had an influence on those empires, naturally.

Unfortunately the website is lacking in painting a comprehensive picture of Persian or those other tribes that formed the Persian empire or perhaps I couldn’t find it.