Connacht were denied the landslide success they at one stage looked to be heading towards but victory over Zebre in Parma was never in doubt.
Tries from Rodney Ah You, Kieran Marmion and Eoin Griffin - plus a penalty try - gave the Irish visitors a 27-19 bonus-point win, with a flurry of late points from Zebre seeing them gain scoreboard respectability.
At one stage early in the second half Connacht were racing away with the game at 27-0 ahead, but their hosts finally came to life.
With both teams struggling towards the foot of the RaboDirect PRO12 table, Connacht came to Italy looking to complete a league double over Zebre, and having also beaten them twice in the Heineken Cup.
Parks missed the conversion but nailed his kick after Marmion worked his way through for Connacht's second try in the 26th minute.
Referee Ian Davies handed Connacht their penalty try from a scrum in the 37th minute, and Parks' extras gave the visitors a 22-0 interval lead.
Griffin kept the points coming when he dotted down after a fine exhibition of running rugby from Connacht in the 56th minute.
Zebre responded, for the sake of avoiding humiliation. Tries from Giovanbattista Venditti and Samuela Vunisa narrowed the deficit to 27-12, with Troy Nathan kicking one conversion but missing the next.
Dries van Schalkwyk added a late third Zebre try, too late for the Italian side to give serious thought to a more consequential fightback.