Quietus could move his court to the East so he can control newly conquered Parthian lands more effectively...
Hmm. Maybe. My feeling, though, is that this is still too early to move the capital away from Rome, and would be a perfect opportunity for an eager dynast to make a play for the purple and get the support of the Senatorial elites. Whether or not the play is successful is almost beside the point: Quietus would need to move back west to consolidate power, and the only-nominally-conquered Parthian lands take advantage of the unrest to break away again.
Maybe he could grant some authority to the Senate to administrate Western Empire except from military matters though... He wouldnt have risked an open rebellion to the German borders after only 50 years from the Batavian Rebellion...
This wouldn't happen. The Senate was well and truly dead as a ruling body by this point, and the last real push for restored Senatorial power was after Caligula's death almost a century earlier. No sane emperor was going to throw power back at them. A co-emperor would be more likely, or even an able Caesar (like Titus during Vespasian's reign).
Ultimately, though, even if Quietus got lucky and everything went his way, he's still going to have a hell of a time keeping effective control over Mesopotamia. The historical record is annoyingly sparse for the second century, but from most accounts, Hadrian was no slouch when it came to strategic matters. That Hadrian felt the need to fall back to traditional borders is, in this light, rather instructive.
By the same token, though, it's tricky to really map out the Trajan's death and Hadrian's acension. It's certainly possible that Plotina, with or without Hadrian's help or approval, forged the adoption documents that put Hadrian in the purple, but it's far from proven. And even without a formal adoption, Hadrian seems to be have been the most likely candidate: he was a favorite of Trajan, and both the army and the Senate accepted him quickly and without any apparent grumbling. Quietus didn't have quite the same credentials, and even if he and his supporters hadn't been suppressed immediately, I'm not sure things would have ultimately shaken out in Quietus's favor, even if it did wind up a messier affair than in OTL.
And the lack of any kind of in-depth historical data on Quietus makes it next to impossible to speculate on what he would have done if he had become emperor. There's no reason to think he wouldn't have withdrawn just as Hadrian did. He's too much of a blank slate to really speculate about.