The Scarlets, winless in Glasgow since 2003, missed out on what would have been a deserved losing bonus point when Parks potted a late penalty to push the margin to eight points.
The respective place-kickers, Parks and Scarlets number 10 Rhys Priestland, missed two shots at the posts apiece during a relatively incident-free opening quarter.
Neither side could stamp their authority on proceedings, with the well-organised defences shutting down most of the attacking forays.
Glasgow suffered some early setbacks as Parks' two misses were followed by a match-ending injury for prop Moray Low.
On 15 minutes, Priestland was narrowly wide with a penalty attempt from inside his own half, and the young stand-off failed to punish Glasgow skipper Alistair Kellock for a tackle infringement - drawing his second effort away to the right of the target.
There was pressure on the Scarlets when Parks launched a probing kick towards the left corner, and Kellock did well to block down a subsequent clearance attempt from visiting scrum half Martin Roberts.
Glasgow were beginning to look the part and after Parks made a slashing break through midfield and Brown kept the move going close to the touchline, Kellock was mauled forward to within metres of the try-line.
Dafydd Jones managed to save his side's bacon under the posts, competing brilliantly for a ground ball and forcing a relieving penalty.
But the deadlock was broken by the Warriors on 29 minutes when big winger Rob Dewey galloped over for his first league try for the club.
Full-back Bernardo Stortoni made the initial incision on the left, the ball was spread out to the right, aided by some good clearing out from the forwards, and a looping pass from Parks put Dewey over in the corner.
Parks added the tricky conversion but the Scarlets bounced back with a converted try, three minutes later.
Daniel Evans came up in the line, taking a perfectly-timed pass from Jonathan Davies and the fleet-footed full-back put the afterburners on to cross the whitewash from 40 metres out.
There were question marks over whether Davies' pass was actually a forward one, however the try stood and went unconverted.
Evans tried his luck with a long range drop goal shot which had the distance but not the accuracy. Still, the Scarlets ended the half in good stead and a penalty from Priestland, after a ruck offence by Chris Cusiter, gave them an 8-7 buffer to take into the break.
Both sides engineered try-scoring chances on the restart. A crossfield kick from Parks was gobbled up by winger Thom Evans but his international colleague Sean Lamont was quick to close him down.
A bustling run by Scarlets hooker Ken Owens deserved more, yet a lofted pass from Priestland was spilled by an unmarked Iestyn Thomas with the line in sight.
Parks and Priestland traded penalty goals, the latter kicking the Scarlets back in front at 11-10 after Brown was spotted coming in at the side of a ruck.
Building for a final quarter push, the Scarlets mauled forward from a lineout in the hosts' 22 and prop Deacon Manu missed a gilt-edged chance to touch down in the left corner.
He looked to have done the hard work as he bounced off a tackle from Dewey, but he agonisingly lost control of the ball as he twisted his way over the try-line.
That was a major turning point in the game and Glasgow - helped by their hard-grafting back row - regained the lead soon after, courtesy of a well-taken drop goal from Parks.
The fly-half's influence was growing and he popped over another penalty on 67 minutes, just moments after Scarlets replacement Simon Easterby was sin-binned for lashing out at Graeme Morrison.
A TMO decision brought Glasgow no joy as they went in search of a clinching try, while the ever ready Stortoni was denied a try by a superb challenge close to his posts by Mark Jones.
The 14-man Scarlets could not hold out though, and when Evans was whistled up for holding onto possession on the deck, a third and final penalty from Parks confirmed a satisfying result for Sean Lineen and his players.
|Glasgow Warriors Score Card|
|Scarlets Score Card|