In the battle of Thermopylae between the Persians and the United Greek Forces, what role did the mighty war elephant play? The Persians threw Immortals and archers at the United Greeks on the narrow cliff-side but the Greco-Persian wars don't describe how or if the war elephants played a decisive role in the major battles.
War Elephants in the west were a military fad that started with Alexander the Great's encounter with them at the battle of Gaugamella. They became popular for a while, but their ineffectiveness for Hannibal at Zama 113 years later spelled the beginning of the end for the fad. The extinction of the Syran and North African species iced it. By the beginning of the Common Era, the Romans were no longer employing them. The Parthans continued to use them for a time, but had to import them all from India. The Ethiopians continued to use them at least until the birth of Mohammed.
Thermopylae was a few generations before all this. There's no record of war elephants being used there, and this is certainly something one would expect a record of had it happened. Elephants are, after all, the prototypical example of "something that cannot possibly be missed, if it is there".
It's tempting to think of war elephants as some kind of super cavalry, but in reality they were far from that. War elephants were unpredictable and hard to control. At times they were as dangerous to your own troops as they were to the enemy. They were primarily a psychological weapon and used as such. You line them up and send them running at the enemy lines. You're not trying to kill people, you're trying to scare them, and against untrained soldiers this had the possibility of routing a unit or sowing disorder. Against well trained armies, however, they were almost always ineffective.
As far as Thermopylae goes, I don't think Xerxes' army included elephants, which would make the point moot, but supposing it had I still find it incredibly unlikely that you could convince an elephant to charge up a narrow mountain pass like that, and against trained soldiers like the Greeks the risks would have outweighed the rewards of trying a strategy like that.