A lot depends on the nature of Crassus's survival. Did he fight and win a successful Parthian campaign, did he meet with significant reverses but escape with his life? Or did he just decide to rein in his ambition and not march east?
If the first, then we may have to assume a more capable Crassus (or just one willing to face the facts on the ground). He will find himself in an enhanced position in roman politics. If scenario two, a lot of his influence will be lost. he may survive politically with the backing of the other triumvirs, maybe even hold onto his eastern provinces, but he is clearly a diminished partner, and will be looking east for vengence, especially if his son dies as in history (not to say he wouldnt intervene, but he will be weaker than his partners). If the third, then we probably have a much cannier Crassus, with less military acumen than his fellow triumvirs (although he might realize this) but a strong base of operations, and is richer than his fellows. Especially if his ambition remains uncurbed, we might somehow wind up with a three-way civil war (hard to manage, but Crassus surviving is a pretty major POD, after all).