I'm just gonna summarize my position as: The rise of the Sassanians forced a paradigm change in the disposition of the Roman army, which before the overthrow of the Parthians had a much smaller deployment in the East. This left a strategic vacuum in the rest of the empire, which gave the Franks, Goths, and Alemanni the opportunity to capitalize on Roman weakness,
Even if you're through talking here, I'd still like to respond to this.

Look at Forum Terebronii. What if that had never happened or, what if the Roman army had prevailed? I think there's a pretty good chance Shapur would've stayed in "his own box"--possibly for the rest of his reign, so the Sassanids would've seemed far less of a threat than in the OTL. The experience of c 233 and 243 CE had, I believe, taught the Sassanids that offensives were futile as long as the Romans could send reinforcements from Europe. When this was no longer possible, after Forum Terebronii, they began an allout assault...
Evidently, Shapur was emboldened by the loss of Danubian fighting power. For the time being, European reinforcements were not available. Now the eastern army had to manage on its own, for some time, without such help. And the result was predictable...
The point is, it wasn't so much the eastern enemy which indirectly messed up the West but, to a degree, the other way around. The western (gothic) enemy deprived the East of a key backup.
and the emperors' attempts to increase the size of the army (which ballooned during this time from around 150,000 to 300,000 troops) resulted in the debasement of currency, exacerbating the crisis. The confluence of these events left the empire unable to effectively fight on three fronts in the long-term, ergo the eventual solution to the crisis was the division of the empire, which left the imperial government more versatile and less vulnerable to disastrous usurpation. That's one interpretation of the crisis, based in part on Peter Heather's, "The Fall of Rome, a New History", but we'll probably never get a completely accurate historiography, given how incomplete our sources are for the time period.
I don't doubt Sassanid Persia was willing to be mote aggressive than Parthia and in some ways was more capable militarily. But I'd say the same was true of the Goths, Franks and Vandals compared to earlier northern enemies. The barbarian onslaughts of midcentury were much more extensive and ambitious (and devastating) than anything the Empire had ever seen before. So while Persia was certainly a factor I wouldn't rate its threat as the principal or most important one by far.