To be fair to the Persian elephants, a large part of the Islamic victory against Persia was not because of any superior skill but because of severe infighting among the Persians. When the Arabs invaded, Persia was literally in the middle of a civil war. Soldiers and generals would betray their commanders midway during battles, and there was often zero coordination.

The Romans on the other hand... lots of hubris and underestimating. There were some brilliant strategies employed by the Arab generals that are even thought in military schools to this day. For example, the time when Khalid bin Walid conquered the walled city of Antioch - without siege weaponry, but by drawing the Romans out by cunning.