What if the Romans had develop Cavalry after Crassus

Yes, you can fight from horseback without stirrups. The advantage is that they let you impart more force to your melee attacks without the attendant risk of being dismounted, especially for the downward blows needed for dealing with infantry, and make it harder to dismount you as well. An added benefit is that they make it easier to mount and dismount as well, so riders with heavy armor and weapons can do so unaided rather than being helped into and out of the saddle. Stirrups let cavalry deliver much more effective shock attacks, which goes a long way toward making them an effective combat force.

Stirrups also make horse archers slightly more effective because they give the rider a steadier seat, which increases his accuracy, but the advantage here is minimal, compared to their effect on melee attacks.
 
Stirrups also make horse archers slightly more effective because they give the rider a steadier seat, which increases his accuracy, but the advantage here is minimal, compared to their effect on melee attacks.

It allows you to put all the weight on the stirrup which is a much stabler position than the saddle itself when shooting, so yes, even for horse archery it's a significant improvement.
 
But let's remember that in the POD, we are talking about the pre-stirrup era.
No one has an advantage in terms of having stirrups. The Parthians and the Sassanians (both pre-stirrup at least not until virtually the end of the latter's run) probably took armored cavalry to its furthest possibility. I believe their Cataphracts had access to larger horses than the Roman and ERE generally did and may of had more formidable armor until the late ERE era.
 
When properly lead the late republican Romans didn't need to radically alter their army composition to defeat the Parthians. Ventidius proved that at Amanus Pass and Cyrrhestica, both crushing losses for Parthia. The trick was to bring a good complement of slingers and face the Parthians on ground of your choosing.

What you absolutely don't do is what Crassus did- chase the Parthians and allow them to wear down your legions before swooping in for the kill.

I think Caesar, had he survived to launch his invasion, would have been more akin to Ventidius than Crassus. The former was, after all, his protégée.
 
Did the PArthians have infantry?

Yes, both melee and missile troops. Neither was as effective as their Roman counterparts, so the Persians relied on their mounted troops to decide battles.

That's not to say the Persian infantry was bad, but that they were simply not the equal of the Romans, who had the best infantry in the world at the time.
 
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