That's not an apt comparison. Raising large numbers of native horse archers requires a lot of cultural and material resources, such as large numbers of people who are raised riding and shooting. For that, you need lots of horses and lots of people who have both horses and the time and inclination to practice horse archery for a good decade or two.
The Athenians were never a horse-centric nomadic people, and did not possess all that many horses. Compared to the Persians and steppe nomads, their cultures of riding, archery, and horse archery were pretty weak. Greek culture at this time doesn't seem to have emphasized archery; sure, you get leftovers of earlier culture, like Mycenean culture, that valued archery, what with Herakles and Philoctetus being great archers, but that time had long since passed.
Fighting as heavy infantry, on the other hand, requires a functioning body, some weapons, and most likely a shield and some kind of decent armor. The Persians already had large infantry armies, and they certainly had the resources to equip a good number of them as heavy infantry. But they didn't.
Possibly it was because the empire allowed subject peoples to fight in their own style, and most of their satrapies focused on skirmishers, light infantry, or cavalry rather than heavy infantry. Perhaps they thought that it wasn't worth equipping heavy infantry just to fight one specific group of enemies whom they thought they could beat anyway. But these excuses are kinda weak. Persian infantry (especially the Immortals) certainly did engage in a lot of close combat where good shields and armor could've helped. The inclinations of non-Persian peoples might not have mattered if the Persian emperors had just better equipped their Immortals or created their own royal corps of armored infantry. And the Greeks' repeated victories against the Persians should have shown that Persian infantry were rather lacking.
Maybe the Persians just didn't suffer enough defeats over a long enough timespan to realize the need for heavy infantry. The later Sassanids had heavy infantry, but their situation was different. The Achaemenid Persian emperors were indisputably the most powerful sovereigns in their part of the world, and possibly in the whole world. They were used to victories and had few major enemies until Alexander. The Sassanids, on the other hand, had to fight the more powerful Roman Empire for over three centuries, over the course of which they gradually mastered siege warfare and got heavy infantry. They weren't as powerful as the Achaemenids, so they may have seen the need to adopt any techniques and technology that could have helped.
But I dunno. Knowing that your lightly- or unarmored footmen with cane shields get massacred by heavy infantry, and then not doing anything about it, seems like trouble.