If Trajan had conquered the various Parthian lands, it is very possible that the Sassanids would still have arisen and taken on Rome (weren't they just the Pars Satrapy rulers during PArthian times?)
But then, alternatively, Rome would have been able to put them down. As they did with most of their regions
It would be horrible over-extension though. how could they manage to keep communications between Cornwall and the Indus river?
> how could they manage to keep communications between Cornwall and the Indus river?
As people used to point out (before classical history fell off the curriculum, first in public school, then in colleges and universities), the US president can communicate to the other side of the world faster than Caesar could communicate across his palace. The borders of the empire were connected with the rest of the empire via the military roads system, which was a great advantage considering no one had a better one.
The main impediment for Rome was in having a non-Roman capital that produced nothing (hmm, sounds kinda familiar) trying to maintain control over rigorously Romanized provinces where most of the armies lived. The breakdown during the 3rd century led to economic activity at levels not previously seen, as there was local/regional control of things, and competing polities kept the various leaders off the backs of the populations they were trying to rule.
That century also saw a crackup of the Chinese empire, the fall of the Kushans in India, and the rise of the Sassanians. If the Sassanians really were such a threat, they had the opportunity of all opportunities to strike Rome — sounds like they had plenty to try to handle in their new conquests.