Avoid the reign of Shapur III, who ruled the Sassanian Empire from 426 to 462 and was often referred to as a second "Darius the Great".
Right, but I think the damage was done long before Shapur, in some ways, and a lot of his success was predicated on the fact that he incorporated the Onnashahs into the empire in the 370s and 380s. The Onnashahs managed to settle in Hyrcania but had a hard time campaigning against hard-bitten Persian troops in the mountains, and ultimately received Hyrcania as a settlement area in exchange for their service to Shapur's forebears.
Maybe if Balamur had led his little chunk of the Xiongnu towards the Danube and continued on west, he would've hit the Western Empire, and you'd have a lot more Latin writers talking about "the Huns." As it is, Shapur could draw on Onnic cavalry during his campaigns because the Onnashahs decided to turn south and go around the Caspian Sea.
The Onnashahs were dangerous enough in the mountains and the desert, and repeatedly bested the eastern legions even in hill country. Imagine what they could do once they got past the Germanic woodlands and had a clear shot at the nice rolling temperate parts of Europa. I imagine they'd displace plenty of other unsettled tribes, too, sort of like the wave of nomads they pushed into the Subcontinent. A massive Hunnic invasion might even push up the
Volkerwanderung by a few centuries; the Germanic peoples really only started to migrate in the seventh and eighth centuries, by which time the Eastern Empire was already fallen.