The Cape outfit won both their opening fixtures on home soil and have now added a first, bonus-point victory on the road in the Northern Hemisphere to their title defence.
Junior Springbok captain Sacha Mngomezulu, normally flyhalf but playing inside centre on this occasion, delivered a polished display and contributed 22 points to the DHL Stormers' deserved triumph via a try, four conversions and three penalty goals.
It was the Cape Town outfit who went into the break 17-7 ahead, having scored two converted tries and a penalty goal, while the hosts replied with a converted try just before the break to ensure they stay within touch.
The visitors eventually scored four tries, four conversions and three penalty goals, while Zebre responded with two converted tries and two successful penalty goals.
The DHL Stormers made a good start when they crossed the line with hardly five minutes of play on the clock lead 7-0. Loose forward Junior Pokomela rounded off a powerful driving maul for the first try of the match which was converted Mngomezulu.
Zebre Parma did their best to disrupt the visiting set piece, however that handed their opponents some penalty opportunities, one of which Mngomezulu slotted to stretch the early lead to 10-0 in the bright Italian sunshine.
The home side stretched the DHL Stormers defence at numerous occasions when they managed to get in possession, unfortunately most of those efforts were thwarted by a combination of poor options and sturdy defence.
Big front ranker Neethling Fouche went over for the DHL Stormers' second try of the game just before halftime when he was on hand to take a pass after Pokomela had breached the defence with a strong run.
The lead was now 17-0 but Zebre hit back almost immediately when they managed to work the ball through a few phases and allowed scrumhalf Chris Cook to dot down. Pivot Tiff Eden added the extras to reduce the halftime deficit to 17-7.
The hosts moved to within three points when they grabbed a second converted try just three minutes after the resumption through prop Ion Neculai to pull the score back to 17-14.
Mngomezulu also had a hand in their third try when he kicked and gathered a loose ball to send Herschel Jantjies over and his third conversion allowed Salmaan Moerat's team more breathing space with the score at 24-14.
With both sides making second half replacements, Eden and Mngomezulu then traded four penalty goals between them to take the score to 30-20 with less than a quarter hour remaining on the clock.
However it was the impressive Mngomezulu who crowned a fine individual performance when he took an inside pass from Manie Libbok to dive over for their bonus-point try to seal a well-deserved away win.
Zebre Parma 20 (7) - Tries: Chris Cook, Ion Neculai. Conversions: Tiff Eden (2). Penalty goals: Eden (2).
DHL Stormers 37 (17) - Tries: Junior Pokomela, Neethling Fouche, Herschel Jantjies, Sacha Mngomezulu. Conversions: Mngomezulu (4).Penalty goals: Mngomezulu (3).