The hosts dominated early on and led 10-0 through Tommaso Boni's try and the kicking of Guglielmo Palazzani.
Ashley Beck crossed to claw Ospreys back in it at the break while Dan Evans scored two quick tries on the return,
Evans touched down again to seal his hat-trick and an Ospreys bonus point before Dafydd Howells' late try.
The result means Ospreys have now won their last six Pro12 games on the bounce while Zebre remain bottom of the table.
Ospreys were without 14 players away on international duty and were given a scare in the first half when the hosts looked comfortable against the in-form Pro12 side.
Zebre, on the other hand, were without 12 players on Six Nations duty with Italy, but centre Boni and back-rower Federico Ruzza started after being released from the Italy squad.
The visitors, who came to Parma on the back of a 12-game winning run in all competitions, had looked like their fine run of form was coming to an end after Boni crossed, but Wayne Pivac's side came out in the second half with an added vigour.
Man of the match Evans scored his ninth try of the season while Sam Parry and Howells also got on the scoresheet to round off a comprehensive victory and keep Zebre scoreless in the second half.
Zebre: Zebre: Kurt Baker, Mattia Bellini, Tommaso Boni, Matteo Pratichetti, Lloys Greeff, Serafin Bordoli, Guglielmo Palazzani; Federico Ruzz, Johan Meyer, Derick Minnie, Valerio Bernabo, Quintin Geldenhuys (captain), Guillermo Roan, Oliviero Fabiani, Bruno Postiglioni.
Replacements: Adriano Daniel, Andrea De Marchi, Bartholomeus Le Roux, Gideon Koegelenberg, Davide Fragnito, Carlo Engelbrecht, Faialaga Afamasaga, Maicol Azzolini.
Ospreys: Dan Evans, Ben John, Kieron Fonotia, Ashley Beck, Dafydd Howells, Luke Price, Tom Habberfield (captain); Paul James, Sam Parry, Rhodri Jones, Adam Beard, Lloyd Ashley, Tyler Ardron, Sam Underhill, Dan Baker.
Replacements: Hugh Gustafson, Gareth Thomas, Daniel Suter, Rory Thornto, Olly Cracknell, Matthew Aubrey, Hanno Dirksen, Jonathan Spratt.
Referee: David Wilkinson (IRFU).
Assistant referees: Sam Grove-White (SRU), Stefano Bolzonella (FIR).
Ospreys back Dan Evans scored a hat-trick of tries as they came from behind to beat Zebre in the Pro12 and keep the pressure on leaders Munster.