It's all a bit more complicated than that, and not made easier by the myth that Rome had poor cavalry forces (not since the Second Punic War).
First off, Caesar could not simply order the development of cavalry. Republican Rome had no military-industrial complex where you put in procurement money and out came stuff. He would have needed a recruitment pool and a tradition of providing this equipment, neither of which were available. And he had no authority to develop it. What he did (and all good Roman general did) was recruit competent cavalry from areas where you could get it. That included Roman citizens, but since most of Italy was poor horse country, it was mostly Gallic, Hispanic and Numidian horse. Crassus could have done the same had he given the matter sufficient thought. He probably could have even had competent heavy cavalry from allies in the region. But even the best army can only do so much to offset poor commanders.
Secondly, the Roman forces in the Augustan and Principate eras had quite competent and dangerous cavalry. There are no end of surviving misconceptions about them, but we know today that a) they wore armour, b) they were used aggressively and c) they could engage hand-to-hand from horseback. The replacement of the javelin by the composite bow and the introduction of ever heavier and more complete armour was a gradual process, but we have equites sagittarii as regulars in the first and contarii and cataphractarii by the second centuries. As early as the Diocletianic army, it is likely that there was no appreciable difference in the equipment of Roman and Persian heavy horse. The Persians may have maintained a superiority of horse breeding stock, which just goes to show how hard it is to switch even with the authority and funds to do so (imperial Rome had something much like a military-industrial complex by AD 50 at the latest)
By the accounts of most people who have ridden Roman style, stirrups are overrated. Personally I've never met a horse I'd trust to go near, let alone get onto, but I'll trust experienced riders over Lynn White, and I have seen a couched-lance attack unseating a 100-kg sandbag ridden from a horse saddled Roman-style. I doubt that was as central an innovation as it is made out to be.
Still, the conclusion I come to looking at the Parthian Wars is that what Crassus needed to emerge victorious was not more or better horses, it was more and better brain cells. The cavalry he could have had was as good as that which Trajan took to Ctesiphon.