Salva sis semidiva Heraklia

Assuming no Civil War, Pompeius would have been the better choice because:

1. He had a secure base of operations in the East from which to conduct a campaign against the Parthians.

2. He a closer relationship with Crassus than had Caesar in terms of senatorial status, and was thereby a more likely choice to avenge Crassus.

3. Pompeius was still an excellent general, and one more suited to dealing with the Parthians. Caesar had never faced an enemy like the Parthians, and would not until his African Campaign. A Roman army sent against the Parthians would need diverse cavalry units, which Pompeius understood and could deploy. Caesar would more likely have fallen into the same kind of trap as did Crassus, just as he was to do in Africa. In spite of the final outcome, Pompeius had out generalled Caesar in the Civil War and was better suited to deal with the Parthians.

4. Without the Civil War Caesar would be pressed more to remain in or near Rome because of his office as pontifex maximus.

5. Also without the Civil War the Senate would have a large faction of Pompeians to contend with. The Senate might be thought to keep Caesar in check to a degree while Pompeius was away.

It would really come down to who the Senate would prefer to remain in Rome. They might think it better to have Caesar go off and meet the same doom as had Crassus, and I think they would be right that Caesar would fail in such a campaign. He would not in that case have had his veteran legions. He did not have experience in employing or combating against cavalry as he would face in Parthia. Also in all of Rome's campaigns into that region, including Julian's campaign, their weakness was a long supply line. In spite of what you may think, this was a real weakness in Caesar's case because it required better organization than Caesar ever displayed. Caesar was central to his political faction. With him gone the Caesarians had little ability to control the political situation in Rome, where as Pompeius was more a figurehead for other political forces and his absence would not have been as critical to their interests. Actually, without the Civil War there may not have been any consideration of a Parthian Campaign. Who to send would cause friction that could lead to Civil War, and then if one was sent it posed Civil War anyway.

There were other campaigns that Caesar might have undertaken if there had not been a Civil War. He could return to Britannia for a conquest, or enter into Germannia. Without the Civil War I don't know if the Senate would have wanted a war with Parthia. Pompeius certainly did not need another expansive campaign to improve his reputation. The focus for both factions would remain in Rome, so distant wars would not have been an option either would likely take.