Trajan was one of the best show-runners Rome ever produced, and was getting set up for the final push to bring Parthia into the empire as a set of provinces. Then he up and died. I’ve viewed that with as much suspicion as the earlier sudden death of the young Alexander the Great.
One of the really inspiring “what-if” scenarios is, Trajan lives, finishes off that last serious long-term enemy of Rome and opens up the direct overland trade routes with China and India (they had sea trade, and with India and points as far east as Sumutra that was quite substantial). He had spent a lot of time and gone to some trouble to conquer Dacia, and clearly didn’t see or wasn’t intimidated by the arbitrary limits some set for expansion.
Of course, we don’t get to know the road not taken, too bad about that. :’(
Parthia as a set of provinces is an interesting theme — I doubt he would have pushed further east of the Indus — too hot. But he could have pushed north and around the Caspian. If the Romans had advance warning of the moves west of the Avars and other Turkic groups, perhaps the entire Germanic migrations may never have happened
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?