How did Trajan defeat the Parthians?

[Robert Vermaat]

Well, the horse archers aren't all that hard to beat, since Roman cavalry typically has armor and shields. They can close to javelin range without many casualties and do bad damage to the unarmored Parthians. Crassus' cavalry at Carrhae did that, they just over-pursued and got themselves cut off and slaughtered. (Oops.)

Not that hard to beat? Well, I doubt that it was ever easy, since (as you may see in RTW), lighter cavalry can never really be caught by not-so-light cavalry. The armour and shields work well in defense, but if you can't close with them you can only drive them off. I don't agree that you can get close without too many casualties, btw. Even if possible, the horse archers would evade, tempting you to go in pursuit. Which always carries the risk that you're cut off from your support infantry. Bad idea.

The cataphracts are a slightly tougher nut to crack, but my guess is that Trajan (and other commanders) simply hired other cataphracts from the eastern areas. Combined with other Roman heavy cavalry, that would probably do the trick.

Armenian cataphracts, always a good idea as support troops, but a steady line of infantry works far better. Cataphracts only work with effect when the infantry is in disarray (softened up by the horse archers is the idea), but they are not a steady force. As Later Roman armies showed (and which Arrian already theorised about) is that heavy infantry, even heavily outnumbered) can hold against vastly superior number of (armoured) cavalry. Even better when they are supported by plenty of missile troops, as the Romans learned after Carrhae. And as Arrian advised, once you have them on the run (they tire easily and can't do much of the surprise wheeling tactic), that's when you let the cavalry loose after them.

Remember that the Roman cavalry always had their infantry as a nice solid line to fall back on to regroup, whereas the Parthian army was entirely mounted. If their cavalry couldn't stop those legionaries cold, the Romans could simply walk right up to and through the Parthian camp, capturing all the baggage, provisions, families, etc. There goes your army.

Indeed, which is why Roman infantry developed into a redoubt for ever growing numbers of cavalry - most likely the Eastern Wars were the cause for that.

Also keep in mind that the Romans could march farther in a day than even a Parthian army. Sure, horses are faster on the battlefield, but a good infantry force can out-march a cavalry force.

Not entirely correct, infantry does not outmarch cavalry in a day, but in three, after when the difference becomes ever larger.