Does this lead to the RE converting to Zoroastrianism instead of Christianity? Or does a Persian brand of Christianity emerge?
The problem is that Zoroastrianism as we know it, developed during the Sassanians, we don't know much how it looked under the Arsacids.
First of all: your best bet is probably a more martially competent and confident Alexander Severus. The Parthians were on their last legs, and Alexander had some very solid resources at his disposal, but seems to have lacked the expertise and confidence to utilize them.
Second: It won't happen. The logistical challenges alone are all but insurmountable. If you take a look at
http://orbis.stanford.edu and run the numbers from Rome to northern Mesopotamia, you'll see that the cost in time and supplies are comparable with the British border, along Hadrian's Wall. And thats the closest part of the Persian Empire, with much of the costs reduced by plenty of maritime transport options. A Roman army occupying Mesopotamia would be at severe disadvantage, far from the Mediterranean transportation network that would keep it fed in most of the rest of the Empire. A Roman army occupying Persia itself would face even longer odds. At least Mesopotamia has its river network.
What you could see is a possible conquest of Mesopotamia, whose main value would probably be in denying its resources to the Persians, rather than being valuable to Rome in and of itself. This could be a long-term situation. Meanwhile, Persia itself is beyond the ability of the Romans to conquer and hold. Perhaps a daring general (hopefully already an Emperor, or else he's going to be reigned in very quickly) would be able to topple an already-weak dynasty, but they're not going to be able to stick around for long. If our Emperor is smart, he'll be happy with a divided Persia. If we go with the idea of striking as the Parthians are falling, then attack the Parthians, and support powerful rulers in the core areas of Media and Persia and Parthia, and hope for a relatively even three-way division of the region. It won't last, it won't be stable, but the best case scenario would be a century or two before one of those rulers, or the Kushans or White Huns, re-unite Persia.
Or, get someone to build an early Suez Canal that can support the logistical strains of a Mesopotamian campaign, and then, Mesopotamia and the Persian coast become much closer. Thats not meant as a serious proposition, but a demonstration of how hard this is.
1. Alexander Severus is too late, the dark clouds were gathering in the north aroud this time. And the fate of Maximianus Thrax shows us that the imperial society wasn't willing to pay higher taxes for stronger army needed to repeal the northern barbarians.
2. This doesn't matter. They won't be bringing their supplies from Rome. There is enough grain in Mesopotamia itself to feed legions, and even money for soldiers' salaries, comes mostly from closer regions and Mesopotamia itself, which are the richest parts of the Oikoumene. For how foraging looked in the Roman Times, I reccomend looking into Ceasar's diaries.
3. Mesopotamia's value isn't in denying it's resources to the Persians (whatever they are, because I'm not sure, we are talking about the Arsacids or the Sassanians, or just about common folk which spoke one of iranian languages in the Plateau), but that together with Egypt that is the richest and the most fertile land in Oikoumene known to the Romans. It was possible to foreigners to conquer the Iranian Plateau in history, so it would have been possible for Roman if they had seriously tryied. I mean, Alexander, later the Parthians (yes, the Arsacid Dynasty were iranized steppe nomads!, and finally the Arabs). Kushans? I reccomend to look at the map of the region. The Iranian Plateau is divided into two parts, by deserts in the middle.
4. The Suez Canal? What doest it has common with the military campaing against the Arsacids? But also there was some canal during the roman period, or slightlhy earlier, I don't remember exactly, but roman traders sailed to India from egyptian ports on the Red Sea.
Gross underestimation of Iran strikes again!
I hope you're aware that Iran is, well, slightly larger than the area between the Rhine and Elbe rivers, as well as slightly more politically organised, slightly more religiously opposed to Greco-Roman annexation, and slightly more densely populated. All those "slightly" are actually "far, far".
None of the Roman conquests were quite as large and developed as Iran - the closest there was to a large-scale integration was Carthage (whose Punic population was concentrated in a single city and thus was easy to break) and Gaul (which was divided amongst sevral tribes and groups). Iran is a whole other thing- there's a reason why it didn't happen, and it's very similar to why China never fully conquered Korea or Japan, for instance - if Korea and Japan were basically the same size as China.
The Seleucid government had to exclusively deal with those areas and Syria (which is a natural territorial configuration - most large Ancient empires that held Persia also held Mesopotamia, and tried to control Syria), not with their own huge empire. Plus, the Seleucids were far more descentralised. And it's not like they had an iron grip over Iran - the Parthians started rebelling in the 250s BC, which isn't 150 years, but rather closer to 70.
I would rather say that "gross understimation of Rome strikes again".
First, we need to establish, a time period about we are talking. I vote for the Principate, because for the Late Empire and the Sassanians I think that we can agree that the Romans were to weak and the Iranians too strong for the roman conquest.
1. During two generations(Cesar in 50s BC and August later), the Romans conquered and estabilished in Europe provinces in lands of Gaul, Noricum, Retia, Panonia, Ilyria, Dalmatia, Moesia. It isn't a small area.
How was the Parthian Kingdom organized? It wasn't that well organized conglomerate of domains. Densely populated? Well, Galia during Ceasar's conquest also wasn't an empty woodland with wisents and auchrons. Mesopotamia was densely populated, but also the easiest for conquest. For the Plateau. Media, Susiana, Persis weren't empty but had their chunck of mountains, deserts and such, and to be honest, it is easier to conquer a densely populated developed land than woodland. Religion didn't matter for the Greeks (or Macedonians, whatever), so didn't matter for Romans. Still the religious situation in earlier period wasn't the same as during the Sassanians. Carthage ? It's a myth. The Romans destroyed just one big city, when other punic/phoenician cities in future Africa Proconsularis were left intact, and phoenician civilization thrived in them to romanization which happened many generations later. Because the Romans weren't interesting in destroying all this land, but in destroying one particular city, which they hated so much. So there weren't any problems with integration of carthaginian lands. Romans conquered and integrated into their empire the whole greek speaking east, ok, that was a process which lasted long generations, but those lands were very very well developed. Egypt during Augustus' reign also wasn't a small and unimportant land. Iran was also divided into several groups. And Mesopotamia was civilizationally a diffrent stuff than the Plateau. To be honest the Plateau wasn't that developed as was the hellenic speaking east, Mesopotamia and Egypt.
For Korea and Japan. Also bad analogy. China was able to conquer and integrate the south of Yangtze, which lands were initialy populated by no-chinese peoples. Japan and Korea were culturally sinicized (a good word?). For Japan there were only two attempts of invading during the Mongol time. But for Korea, those kingdoms were chinese vassals during various times in their history. There were also a period when area at least of contemporary North Korea was under full chinese control. "Four commanderies of Han".
The Seleucids. Hmmm. Decentralized? Romans weren't decentralized? The Roman Empire under the principate was also very decentralized. It was basically just two levels. The Emperor with his chanchelary, and the provincial governors, outside that there were countless towns and cities, which governed themselves and collected taxes for the Emperor, in places not yet urbanized there were "tribes/peoeples, etc". Urbanization in roman termst I mean when there is formed something which governs themselves like polis, not just a town in functional meaning, because bigger gallic opidia were towns in functional meaning, but not in roman meaning. Still the most important thing, the West of Plateau vs. the East of Plateau. It wasn't that easy to cross it, and there weren't that many ways. So nothing strange that the Seleucid after they lost Parthia proper for the Arsacids circa 250bc didn't reconquer that, but still the kept the western part, Susiana, Persis and Media. Which were for them far more important than some remote area behind the Caspian Sea and Hyrkania. So that's 150 years of control of the West of Plateau. Also those parts were the most important for Romans, not some semilegendary Bactria or Ariana.
As I say it is important what we want Romans to conquer, the whole plateau, the west of plateau, or Mesopotamia. All scenarios are diffrent.