After the scrutiny and controversy that surrounded last weekend's first game against Israel, the Republic of Ireland returned to Dublin for their first competitive outing on home soil since beating Portugal 2-0 in a World Cup qualifier 11 months ago.
Parrott scored both goals in that game and again had his name up in the Aviva lights as he bamboozled the Austrian defence to open the scoring in superb style after 12 minutes.
Having picked the ball up just outside the centre circle, Parrott poked the ball through Philipp Lienhart's legs and skipped past David Affengruber before coolly slotting a right-footed effort under goalkeeper Florian Wiegele.
Like in the Israel game, the Republic of Ireland looked sharp going forward with Hallgrimsson opting for an attack-minded line-up that included the returning Finn Azaz in a deeper midfield role behind Parrott, Adam Idah and Jack Moylan.
Before the game, Hallgrimsson said the past week has been "difficult" for Azaz, who has family connections to Israel and was an unused substitute in Sunday's game in Debrecen.
Goalkeeper Gavin Bazunu, meanwhile, returned to the bench after making himself unavailable for both Israel matches.
Azaz was involved in the build-up to the hosts' second goal on Thursday, his pass to Ryan Manning setting up the wing-back's cross that caused havoc in the Austrian defence before Manning was fouled by Nicolas Seiwald.
After Parrott confidently drove the ball home from 12 yards to double the lead, the hosts looked on course for another impressive victory but Austria had other ideas.