I am from Iran and I saw just a few episodes. This story was famous in my parents' childhood and I really wonder why it is so different from the original story (the author is called Naghibolmamaulek). And why is the theme completely different from Persia?
The anime's based on Hiromu Arakawa's manga of the same name, which is, in turn, based on Yoshiki Tanaka's fantasy novel series of the same name. As the Wikipedia page for the latter states, it appears to have taken inspiration from a few different sources:
Whilst the protagonist's name (which is Turkic for lion) appears to may have been taken from the popular Persian epic of Amir Arsalan, other than this anachronism, Arslan and his Parsian enemies and allies primarily share many parallels with Cyrus the Great and other historical figures of 6th century BCE Persia (albeit with several liberties taken), whereas the conflicts with the Lusitanian forces (which bear the Byzantine Orthodox cross) – despite mostly French names and a certain religious zealotry implying a connection to the (Catholic) Crusades (again, with liberties taken)– appear to be based on the Byzantine–Persian Wars, specifically those of the 6th century CE. Furthermore, several names of prominent Parsian characters appear to be taken from known important figures throughout Persian history as well as the historically unsubstantiated legendary parts of the historiographic Persian epic Shahnameh. Additionally, supernatural elements mostly based on ancient Near East mythology increasingly play a role as the series goes on.
I wasn't able to find any reference to Naghibolmamaulek, the author you mention, but the short answer to your question is: the anime's based on a story that was inspired by the one your parents may have heard during their childhood, among others, and it is not a direct adaptation of it/them. As such, Tanaka (and possibly Arakawa too) took some liberties and changed a few things around.