OK, when looking at this question, I will not focus on the battle itself as a general overview of it can be found in the Iranica article and wikipedia. Rather, I will focus on the Parthian military and its traditions as well as why the battle was such an impressive victory.

To start with, many historians have written off the battle as a fluke. Being that Crassus was well, an idiot, his mistakes are the reason the Romans lost, and if there were to be a proper Roman general (Caesar, Pompey) the results would have been much different. There is a huge problem with this idea because the battle is a Roman defeat, not a Parthian victory. The subtle difference being that little attention is paid to what general Surena actually pulled off: with an army of merely 10,000 men defeated an army of 40,000 Romans. Let me say that one more time: Surena with an army of 10,000 men defeated an army of 40,000 Romans. Yet this is almost never mentioned in historiography; generals would never fight being that terribly outnumbered, and not only did Surena fight, he won.

This arises for a few reason, primarily that Surena was a military genius and carefully used shock tactics as well as effectively using his nomadic force to harass and make life difficult for the Romans. For the most part, the army consisted of roughly 1000 heavy cataphract cavalry augmented with around 9000 lighter horse archers. This army was not too different from other nomadic based armies. Historically, the Parthians had many enemies, ranging from the Successor armies of the Seleukids and Baktrians, to the terrifying battles against the nomadic armies of the Yuezhi and Saka. The Parthians had seen it all, from infantry based armies to heavy horse armies mirroring their own. And not only did the Parthians fight these armies, but they defeated all of them. The Seleukids attempted an anabasis under Antiochos VII Sidetes, and despite three crushing victories, their current king Phraates II Nikephorou defeated him outside Ekbatana. His predecessor Mithridates I Philhellenos spanked the Baktrians under Eukratides so hard, that within a few years, nomadic tribes invaded and destroyed the Baktrian kingdom. And then when the Saka were causing problems in the eastern border of the empire, Gondophares of the Suren clan pursued them into India and formed an Indo-Parthian kingdom.

I bring up all these factors to emphasize the rich history of the Parthian military as they truly had fought almost every kind of army imaginable. Unfortunately, in the veneer of the conflicts with the Romans, this is ignored and emphasis is taken to suggest the superior Roman legionaries countering the Parthian cataphracts. And to further emphasize, the Parthians in their history would fight more terrifying armies than the Romans. The battles between them and nomadic tribes in the east were likely incredibly violent owing to the overall similar tactical lineup. And that really is the story for the Parthians: they were surrounded by superpowers (the Romans never were that unlucky) and yet not only survived, but thrived.

With that, we turn to the battle of Carrhae. The previous reason that Crassus sought to emulate either Alexander, or at least gain equal military footing with Caesar must be discarded. Dr. Rahim Shayegan has looked into the campaign, and suggests that Crassus sought to enhance Roman interests in the east by dislodging Parthian client kings and replacing them with pro-Roman ones. Being that Seleukia had recently rebelled, he also wished to set up an ally there. For the Parthians, while they were watching the Romans closely, had little interest in the west as of now. The king Orodes II was a very shrewd king who realized the importance of keeping the many clans in line. So when Crassus entered Parthian territory, he was none too pleased because there was no apparent justification for the war. Surena was dispatched to take care of this army, and we all know what happened after that.

The tactical differences between previous eastern enemies of the Romans must be noted. Both Armenia and Pontos, while relying on a heavy cavalry component, nonetheless focused on their infantry, resulting in a mixed troop. The Romans had no real trouble with this system as both nations were defeated. In contrast, the Parthians, being nomadic in origin, relied more on their cavalry, and while they did have heavy infantry, they primarily used them for garrison duty rather than offensive weapons. These cavalry were delineated into the horse archers and the far fewer heavy cataphracts. The Romans had never encountered this type of army before, and though the Successors (particularly the Seleukids) employed a significant amount of heavy horse, it was often their infantry that would take care of business.

With this kind of army, the Romans were unprepared and suffered a terrible loss. And think historically: the Romans from Sulla, Marius, Pompey, and Lucullus had been running a train through the eastern med defeating kings left and right. And yet at Carrhae, the Romans had actually met a military equal for the first time since Carthage. I think Justin says it best:

""Being assailed by the Romans, also, in three wars, under the conduct of the greatest generals, and at the most flourishing period of the republic, they alone, of all nations, were not only a match for them, but came off victorious;" -Justin 41.1.

And that is a Roman author commenting on the Parthians!

To counter the Parthians, the Romans began to adopt their own cataphract regiments as the best way to challenge a cavalry army is to have your own cavalry. Antiochos III Megas of the Seleukids had realized this on his own anabasis against the Parthians in the 210s BC and the Romans would as well.

As a final point, I would hope that this only answered your question, but also broadened your knowledge about the Parthians. In historiography, they are often forgotten or ignored and this is not fair to a powerful empire. Especially when being compared to the Romans, some characterize them as an eastern despotic empire. Such notions need to be critiqued and we need to take a newer, critical look at the Parthians.

If you have any other questions, please let me know!

Sources used:

Primary--

Epitome of Pompeius Trogus, Justin

Life of Crassus, Plutarchos

Geographia, Strabo

Secondary: Arsakids and Sassanians, Rahim Shayegan

Shadows of the Desert:Persia at War, Kaveh Farrokh

Assorted articles from the Encylopedia Iranica