Julius Caesar was one of the greatest generals in history, and arguably Rome's best general (with the possible exception of Scipio Africanus). In addition, as evidenced by his visit to the tomb of Alexander the Great, he idolized Alexander, and I am sure would have loved to try to do what Alexander did in conquering Persia. Years later Trajan did figure out how to invade the Parthian capital city of Ctesiphon by moving columns down the Euphrates - so a successful invasion was possible.
With Julius Caesar's assassination Rome fell into civil war. Mark Antony tried an invasion, but as history showed, he was not nearly the general Julius Caesar was. Julius's successor, Augustus, was not a military genius in any sense, and was too worried about his position in Rome to risk an invasion of Parthia, and was content just to get the Roman standards back that had been lost by Crassus.
Subsequent history may have been a lot different if Julius Caesar had not been assassinated when he was.