The "static" style of horse-archery (I've seen it called "Iranian") was pretty common AFAIK, and stemmed from entirely practical considerations. Unlike the steppe nomads with their herds of little tough horses the warriors of sedentary populaces had to make do with just a few but bigger and stronger dedicated warhorses, and that meant they couldn't afford to dart all over the place skirmishing quite simply because that'd have tired out the mounts right fast and they didn't have a large pool of remounts available. Ergo, static firing was preferred.
As an added bonus that gave better accurate range due to no interference from a moving mount, or so I've read.
The Romans, Byzantines, Arabs and Chinese used the same method for the exact same reason, and that's just the ones I can think of on the fly. Given the economic and tactical considerations inolved with horse barding I'd also guess nomadic elite heavy cavalry had the same approach.
Given that the Sassanids were once subjects of the Parthian realm their aristocracy doubtless had been fighting for quite a while as cataphracts for their masters; the cataphract pattern became the norm for aristocratic heavy cavalry in the whole Middle East following the Seleucid collapse AFAIK (I've seen the Palmyrene catas being specifically mentioned as being of the Parthian model for example). Plus of course the Sassanids needed a fair force of heavy cavalry to begin with to pull off their takeover in the first place - once it became clear which way the tide was turning obviously many a lord of the Parthian empire cheerfully jumped ship to the rebels in the time-honored tradition of feudal barons looking out for Number One. There was apparently a similar mass defection of Sassanid manpower to the Muslims later, following the same logic of abandoning a sinking ship.I've read a version that indicates that the vast majority of Sassanids cavalry were 'static rapid firing horse archers', some with horse barding, and that the Cataphracts found in their ranks came from the Parthian families that were assimilatated after the Sassanid conquest? Sounds fair to me.
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