This scenario can be broken down into four constituent themes:
How might Crassus have "won" at the Battle Of Carrhae? Let's assume that history follows more or less the same course as it did up until maybe the couple days or so prior to the battle. This means that Crassus has already decided on a direct route marching across the desert (rather than a more circuitous route through the mountains of Armenia), and has politely declined Armenian aid (which amounted to about 16,000 cavalry and 30,000 infantry). Thus, the forces available at Crassus's disposal are as follows: about 40,000 heavy infantry, 5,000 light infantry, and 5,000 cavalry. The Parthian forces are: about 10,000 horse archers and 1,000 heavy cavalry. The terrain is a flat desert.
One more note: I put "won" in quotation marks because I don't think Crassus actually needs to win an outright decisive victory in this scenario, he just needs to ensure that his army doesn't get completely routed like it did in OTL and he doesn't get himself killed. I think even a "tactical defeat" would be an acceptable result as long as Crassus is able to withdraw his army in decent order.
How would Crassus continue his campaign, assuming the above result at Carrhae? Would he be able to continue his invasion of Parthia, or would he be forced to withdraw to friendly territory to lick his wounds? If the latter, how would he adapt / reform his army to better respond to the threats he faced at Carrhae? Would he accept aid from the Armenians?
How does Crassus's survival affect the political situation in Rome? Was the civil war between the Caesarian and Pompeian factions inevitable, or is there any hope that Crassus can continue to mediate between the two and prevent (or at least delay) the war? If civil war does break out, which side does Crassus take, and what role would the wealth obtained from plundering Parthia, as well as possessing several veteran legions, play in the ensuing conflict?
Is a Roman annexation of Parthia now possible? Would a military defeat of this magnitude be enough to spell the downfall of the Arsacid dynasty? How would the Romans rule a hypothetical Persian province? And what impacts does this produce down the line regarding the Silk Road, Roman contacts with China, and the rise of Islam?