If the necromancy be excused, Anthony Clipsom made me aware of this article on Achaemenid cavalry: All the King's Horse
Some observations of relevance for this thread:
Regarding bow use, Tuplin observes that bows are depicted in the hands of Persian horsemen (more commonly hunters than cavalrymen, apparently) on seals etc, but much less often than spears, and mostly in Xerxes I's reign or earlier.
Regarding shields, assyriologists have apparently reinterpreted Gadal-Iama's two shields as a quiver and a scabbard (and redated the document to 421 BC). This leaves the man with somewhere to put his 120 arrows - but no explicitly mentioned bow!
In another inscription, Darius is more explicit about shooting from horseback:
I am skilled both in hands and in feet. As a horseman, I am a good horseman. As a bowman, I am a good bowman, both on foot and on horseback. As a spearman, I am a good spearman, both on foot and on horseback.
(whole inscription reproduced, transcribed, and translated here
Acc'd Tuplin's summary of Greek accounts - such Persian and Babylonian battle accounts as exist have no tactical detail at all - Achaemenid horsemen are most commonly described as fighting hand to hand with spears or swords; less commonly as throwing spears, and less commonly still as shooting with bows. He finds little reason to assume that there existed different classes of cavalry specialized in different weapons or tactics, but acknowledges as plausible that Saka et sim.
contingents may have been especially proficient in horse archery. This presumably rules out Horse Bow for the bulk of Achaemenid cavalry.
He's skeptical there was much practical difference between the cavalry of Darius I and Darius III.