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.