Glasgow shook off their jitters to step up their stirring revival with a precious bonus-point triumph over the Irish dark horses in the Magners League.
But the Warriors were forced to survive a late rally and a spate of bizarre refereeing verdicts by erratic Welshman Hugh Watkins — which reduced them to 13 men — before sealing this crucial success.
Glasgow had made a shaky start as the visitors drew first blood with an opportunist drop-goal by Mark McHugh.
Connacht doubled their tally with a straightforward McHugh penalty from 30 metres.
Fit-again Argentinian Francisco Leonelli then burst his way into the match by grabbing the opening try.
Chasing his own hack towards the line, Leonelli appeared to be only third favourite to reach it first. But the Connacht markers failed to mop up the peril — and sheer determination enabled the full-back to pounce and flop over.
The province bounced back immediately to create an excellent score for their number 15, Dan Riordan.
And the visitors maintained their patience in the face of tigerish tackling in midfield, with Riordan making the crucial break after latching on to a perfectly-weighted pass from Darren Yapp. McHugh slotted the conversion.
There was a double blow for Glasgow as McHugh followed up with his second penalty for failing to vacate the tackle scene — and Parks was out of luck again at the other end.
Parks' fortunes took an upward turn two minutes before half-time when he hurtled in for a close-range try and added the extras to leave his side adrift by a single point.
Within seconds of the restart, the Warriors powered back into the lead when skipper Ally Kellock found an extra surge of pace to reach the line at the end of a 20-metre rumble.
It took Glasgow just two more minutes to make sure of the bonus point, John Barclay providing the finishing touch to a slick move. Parks added both tricky conversions for good measure.
And centre Graeme Morrison snuffed out any prospect of a Connacht fightback with a superb solo score, leaving three defenders in his slipstream. Parks again chipped the goal.
A mixture of slackness among the back ranks and puzzling refereeing by Watkins allowed the visitors to put a flattering complexion on the scoreline.
Ray Ofisa and Brendan O'Connor grabbed tries, the second of which came after Warriors centre Andy Henderson was the victim of a strange yellow card verdict.
Watkins then infuriated the crowd even more when he sin-binned Scott Barrow for obstruction when a Connacht player appeared to deliberately run into him. Paul Warwick kicked the penalty to set up a nervy finale.
Parks' penalty steadied the Glasgow ship — only for Yapp to snap up the touchdown which captured two consolation points for the Irish side.