The news of the attack would reach you via a government courier (think pony express) who travelled along the Cursus Publicus (the public way). These paved highways connected the empire to its distant provincial territories and trading partners. It was created by the Emperor Augustus to facilitate the flow of taxes, messages, and services. Take a look at the Tabula Peutingeriana to get an idea of what the extant routes were in the year 400AD.

Here is a description by 7th century Procopius describing an earlier time:

"The earlier Emperors, in order to obtain information as quickly as possible regarding the movements of the enemy in any quarter, sedition, unforeseen accidents in individual cities, and the actions of the governors or other persons in all parts of the Empire, and also in order that the annual tributes might be sent up without danger or delay, had established a rapid service of public couriers throughout their dominion according to the following system. As a day’s journey for an active man they fixed eight ‘stages,’ or sometimes fewer, but as a general rule not less than five. In every stage there were forty horses and a number of grooms in proportion. The couriers appointed for the work, by making use of relays of excellent horses, when engaged in the duties I have mentioned, often covered in a singly day, by this means, as great a distance as they would otherwise have covered in ten."

Using this description and mapping out the way stations along each route, the scholar A.M. Ramsey was able to determine an average distance of 50 miles per day for a fast traveling system of royal couriers. Keep in mind though that bad news would have travelled faster than good news.

The Iranian borderlands are roughly 2000 miles from Rome, so 50-60 miles per day it would take just over a month to reach Rome.

As an aside were there any Persian empires at the time of imperial Rome to fight? My understanding was that the successor states were Hellenic like the Ptolemys in Egypt and not very Persian.