I came for a fap but as a history nerd and iranian myself I ended up enjoying watching you argue more lol.
Most probably they would have referred to themselves as members of "The Kingdom", and not specifically as Persians either, unless they were from that particular area of the Empire (around Persis itself).
They referred to the Empire as the Achaemenid Empire ("Shahanshahe Haxamenishy" or "Haxamenishah" in persian). The word Iranian ("Aryana") first came up during that time describing the Aryan group of people. Aryan was also the word used to describe us until Hitler slapped a bad reputation onto it, now it's "Iranic" in english.
After the Achaemenid Empire however when rulers changed and the people didn't want to change the name of their empire everytime, so the name for land was fixed to Eranshahr (these words are still used to this day in some dialects or context btw) as it was more convenient. Eran = Iran, Shahr = city/country/land. So basically it says "Iranian land".
So to give the best educated guess they probably called themselves either "Haxamenish" (=Achaemenids) due to their empire or "Aryana" (= Iranian) or something similar due to ethnic background. The word Pars (=persian) is a subgroup of Aryana and as a multi ethnic empire probably not used in this context.
Edit: For the empire I'd also say Haxamenish or Iran. Definitely not Persia, that's beyond dispute.
Thanks for listening to my Tedtalk. Have a good fap.