Having survived the early storm it was the visitors who edged in front with a crisply struck penalty from forty metres by outside half Ceiron Thomas for offside.
Glasgow returned to attacking mode and yet more excellent tackling stopped their surge. Some illegal handling on the deck gave Parks a simple opportunity to level the issue; he was never going to miss from almost underneath the crossbar.
The nervous sparring continued with both teams making mistakes galore and Thomas replied with his second accurate kick from closer range to restore the Scarlets' advantage.
Even better was to come for the Scarlets in the seventeenth minute when they snapped up an opportunist touchdown. Matthew Watkins made the most of slack marking to pave the way for number eight Dafydd Jones to plough over from eight metres out.
Despite having the bulk of possession and territory, Glasgow found themselves further adrift when Thomas confidently completed his penalty treble. At this stage, the Scots were looking strangely lethargic and a shadow of the unit that had performed so well in recent outings.
Having already lost Test loose forward Jonny Petrie to injury at the midway stage of the half, Glasgow were also forced to replace limping winger Thom Evans with Colin Gregor.
Glasgow stepped up the pace in the approach to the interval and went within a fingertip of bagging a try. Fullback Rory Lamont sprinted forty metres before sending a chip across the goal line. Hefin O'Hare was the chief chaser but he lost out to Dafydd James in the race to get the first touch.
The near miss served to inspire Glasgow and just two minutes later O'Hare was in the right place at the right time to squeeze over in the opposite corner following more slick play from Lamont.
Glasgow kept up the momentum after the restart and substitute scrum half Sam Pinder almost wriggled across the line with his first touch of the ball, having taken a tapped penalty. Scarlets captain James Bater was yellow carded for a professional foul and seconds later another of the home bench-men, Stevie Swindall, powered over for their second score. Parks made light of the wide angle to add the extra points.
The Scarlets hit back in positive fashion with centre Regan King being hauled down just a couple of paces short of the target. Parks was narrowly wide with a 50-metre penalty at the other end as Glasgow tried to decisively snuff out the Welsh challenge.
They got their reward when Swindall completed his double, capitalising on excellent work by Pinder, Parks and John Barclay. Parks was off the mark with the conversion bid, but made amends with a penalty to put the Scarlets out of the reckoning for a consolation point.
There was still time for prop Kevin Tkachuk to claim the bonus score, goaled by Parks.
Glasgow 30 Llanelli Scarlets 14
Glasgow: (8) 30
Tries: O'Hare, Swindall (2), Tkachuk
Cons: Parks (2)
Pens: Parks (2)
Scarlets: (14) 14
Try: D Jones
Pens: C Thomas (3)