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The Welsh region had the game wrapped up at half-time after tries from Parry, Lloyd Ashley and Ashley Beck, with Sam Davies adding the extras.
Tommaso d'Apice struck back before the break, before Parry scored his third rolling maul try following a line-out.
Ben John and Dafydd Howells added further scores for the Pro12 leaders.
Fly-half Davies scored Ospreys' first points of the season with a penalty after Zebre were penalised for not rolling away after just five minutes.
Man-of-the-match Parry bundled over from a rolling maul shortly after following a period of sustained pressure from Steve Tandy's side.
And the scoreline became even rosier for Ospreys when second-row Ashley broke through a gaping hole in the visitors' defence with a charge from the 22-metre line.
Centre Beck was next to touch down when he crashed through two tackles from close range on 25 minutes.
Zebre reduced the deficit when d'Apice rumbled over from a line-out five minutes before half-time, but their good work was soon undone.
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Parry forced his way over for his second try on the stroke of half-time with Zebre again unable to stop Ospreys' rolling maul.
It became a night to remember for the 24-year-old when he scored a near replica of his second try seven minutes into the second half.
Parry was replaced midway through the second period, but the onslaught continued with replacement John running in from a fine inside pass from Davies.
Fellow substitute Howells collected a cross-kick from full-back Dan Evans to score with five minutes remaining, before John scored his second after an off-load from Wales flanker Justin Tipuric.
Lock Bradley Davies made his Ospreys debut off the bench after joining from Wasps, while Tipuric started after recovering from concussion.
Tipuric returned during pre-season having not played since landing heavily at a line-out against Italy in the 2016 Six Nations.
Ospreys: Dan Evans; Jeff Hassler, Ashley Beck, Josh Matavesi; Eli Walker, Sam Davies, Tom Habberfield (capt); Nicky Smith, Sam Parry, Dmitri Arhip, Lloyd Ashley, Rory Thornton, Olly Cracknell, Justin Tipuric, Tyler Ardron.
Replacements: Hugh Gustafson, Paul James, Ma'afu Fia, Bradley Davies, Dan Baker, Rhys Webb, Ben John, Dafydd Howells.
Zebre: Kurt Baker; Lloyd Greeff, Giulio Bisegni, Tommaso Castello, Gabriele Di Giulio; Edoardo Padovani,  Guglielmo Palazzani; Andrea Lovotti, Tommaso D'Apice, Pietro Ceccarelli,  Federico Ruzza, George Biagi (capt), Jacopo Sarto, Maxime Mbanda, Andries Van Schalkwyk.
Replacements: Carlo Festuccia, Andrea De Marchi, Dario Chistolini, Valerio Bernabo, Giovanni Pettinelli,  Fabio Semenzato, Carlo Canna, Tommaso Boni.
Referee: Andrew Brace (IRFU)
Assistant referees: Stuart Gaffikin (IRFU), Gareth Newman (WRU)
Citing commissioner: Aurwel Morgan (WRU)
TMO: Simon McDowell (IRFU)

They will perform in Glasgow during the five-day "pop-up festival", as part of BBC Music Day.
More than 60 performers will take part in the free event, which will feature both established and upcoming artists.
The festival, which aims to bring people together across generations and communities through their love of music. will run from 4-8 June.
Artists including Stevie McCrorie, James Bay, Jamie Cullum and Twin Atlantic will also take part, with all BBC Radio Scotland music shows involved.
Sharon Mair, project executive of BBC Music at the Quay, said: "We know that our audiences loved the BBC at the Quay site last year during the Commonwealth Games and we are bringing this five day festival back to celebrate BBC Music Day.
"Glasgow is the UNESCO city of Music and what a brilliant way for us to showcase the city and the fantastic musical talent that will be at the event."
Hooker Sam Parry scored a hat-trick as Ospreys made a winning start in the Pro12 with an eight-try bonus-point win over Italian side Zebre.
[NEXT_CONCEPT]
Lulu, Deacon Blue and KT Tunstall have joined the line-up of artists taking part in BBC Music at the Quay.