The hosts meanwhile remain bottom, but had returning Six Nations star Giovanbattista Venditti on the scoresheet as well as South African Dries Van Schalkwyk.
But it was Liam Williams, returning from Wales Six Nations duty, who proved the difference, with his namesake Owen crucially keeping the scoreboard ticking over which helped prevent a repeat of their nerve-racking 22-13 triumph in the reverse fixture at Parc y Scarlets in November.
Zebre began brightly though, dominating territory, and got their rewards when No.8 Van Schalkwyk stretched out and touched down under pressure from three Scarlets tacklers.
That 13th-minute lead wasn't extended however, Paolo Buso missed the conversion for the Italians.
Scarlets' first points on the board came from the boot of Owen Williams - captain Rob McCusker won a penalty for the visitors and the fly-half duly converted.
In a fast-paced opening to the game Mauro Bergamasco was forced off and replaced by Filippo Cristiano, before Owen Williams notched another three-pointer just after the half hour mark.
The hosts were still pressing and they could have gone into the lead had Buso converted a 38th-minute penalty, but the Welsh side responded with seconds remaining in the half.
Wing Liam Williams crossed in the corner despite the challenge of Sinoti Sinoti and, after checking with the TMO, referee Peter Fitzgibbon gave the try - Owen Williams missed the conversion but the Scarlets still went into the interval with an 11-5 advantage.
The visitors began to pull away after the break, Owen Williams kicked his third penalty of the evening in Parma on 45 minutes.
Then, just two minutes later, they all but ended the contest when lock Snyman scored a sensational try, sprinting from well inside his own half and beating two men before crashing under the posts.
Owen Williams converted with ease then added another penalty to give the Scarlets an unassailable 24-5 lead.
But Zebre are made of stern stuff, and battling for pride Venditti, playing at centre, led the way forward for the Italians with their second try just before the hour mark, although Alberto Chiesa missed the conversion.
Aaron Shingler, who like Liam Williams was making his first start since returning as a Six Nations winner with Wales, dampened his evening somewhat when he received a late yellow card but it made little difference as Simon Easterby's men saw out the remainder of the game.
|Zebre Score Card|
|Dries van Schalkwyk||1||5|
|Scarlets Score Card|