I know Augustus was a good organiser and talented manager but I really doubt Augustus could bring legions from the west to bear on Parthia, that would be very dangerous for the empires security, the disaster at the Teutoburg forest took place during Augustus's reign. In fact fighting along the Danube and Rhine rivers never stopped. The only reason more conflicts like the Cimbrian war never took place was because the legions were on the border, if removed, you'd have full scale barbarian invasion on your hands.
Second assuming there was no threat from Germanic tribes across the borders which I consider unlikely, the legions still couldn't be moved for fear of internal rebellion, if there were no legions, there was practically nothing stopping the local tribes from revolt or local governors vying for the throne.
Now dealing with the actual campaign itself there are a number of problems which the Romans would run into regardless of how good the commander leading them is. The first being logistics. The Parthians repeatedly utilized a scorched earth policy, basically forcing the Romans to depend on supply trains, which the deeper they penetrated into Persia the longer they would become, and the longer they would become the more vulnerable they would become, Antony's baggage train was repeatedly raided, and two legions were taken as prisoners of war.
Another problem the scorched earth policy brought was how big a an army the Romans could bring to field, the bigger the army they fielded the bigger the supply train, from what I know the 100,000 troops Antony took were already large by Roman standards, increasing the size of the army would be unfeasible, it would be expensive, and have dangerous supply issues. Then the problem of the route, Mesopotamia was usually avoided in favor of Armenia who's hilly terrain negated the Parthian cavalry's mobility. Beyond Armenia though the terrain generally became more open, and at this point they would be deep into Persian territory with very long supply lines.
Then there were issues like who would lead the army? Any general given command could easily proclaim himself emperor and march on Rome itself, leading to civil war, and if Augustus personally lead the army then the west would be unprotected.
Trajans solution was to take ships town the Euphrates which was successful, but beyond Mesopotamia he'd face the same problems other Roman generals would face, and its important to remember he ruled during a stable and prosperous period of Roman history, even then his successor Hadrian gave up Mesopotamia and Armenia to The Parthians because they were simply to expensive and too far away to hold. For these reasons I think Augustus couldn't have threatened Parthia with a major invasion, from what ive read about him he was very averse to war with the Parthians and vice versa, the return of the eagles was a goodwill gesture as part of a peace agreement that was formed on equal terms.
