Logistics might be an issue here... As well as a lack of the carruca, the main reason Rome stopped at Germany. There would also need to be administrative reforms.
Not to say it would be impossible - the Chinese have managed to create states of a similar size, though in the case of China, it's core territory is a lot more compact...
Motivation is another aspect; the best comparison here would be Russia, whose primary expansion into Siberia was motivated by, well, not wanting to be raided by steppe nomads for all eternity. Rome was similar, always attempting to pacify various barbarians by expansion.
This gives us a rough timetable for this, as well as the prerequisite conditions: you want to keep Rome powerful until the Hunnic invasions and the start of the migration period. And by this, I mean, you need to keep the Roman bureaucracy and administration strong - it wasn't the mass migrations of various barbarians that ended Rome, but the increasing radio silence from Rome leading to the puppet barbarian lords to just do whatever they want. Now, after the Huns do their thing, it's absolutely critical to procure the carruca - seeing that it was first invented in China, and that the first Europeans to acquire it were the Slavs during the migration period, we can assume that the Huns introduced this technology.
This way you kill 2 birds with 1 stone - Rome now has the technology to properly utilize any further annexations in Europe, and Rome now has a strong motivation to expand further - restoring trade routes, and pacifying the region. If Rome remains strong at this point, it's very likely that some peoples will even *invite* the Romans to rule over them (similarly to how the ancient east Slavs invited the Varangians as a "neutral" party, as no tribe and kingdom managed to overpower the others).
The problem here: Europe has a far higher potential than the core of Rome. The soil is better and allows for a far higher population than the mountainous and arid Mediterranean basin. This Rome will quickly stop being Rome. Eventually, the elite will simply become mostly Germanics and Slavs (basically what happened with the Frankish Empire and the HRE). Latin will be quite an odd language...
Then, and only then, can any invasion of Persia be considered. And it's not going to happen from the Caucasus and Anatolia as usual, no - when you own most of Europe, you own the core of Russia. And when you own the core of Russia, you're automatically implored to cross the Urals and keep going for the same reason Rome would go out their way to annex most of Europe. The invasion of Persia will be from 2 directions: from Central Asia, and from the West.
A state of this ludicrous size and eventual population will probably not be very centralized - instead of an Eastern and Western empire you're probably going to get at least 5.