George Naoupu added a second try, which Parks converted before Tom Prydie's penalty opened the scoring for the visitors just before the break.
Danie Poolman went over early in the second period to all but wrap up the victory, despite a second Prydie penalty, with Parks also landing a difficult drop-goal.
Jonathan Evans went over in the closing stages to salvage some pride for the Welsh region, but Connacht ran out easy winners.
The hosts came into the game after four consecutive RaboDirect PRO12 defeats but started brightly, Parks kicking an early penalty to put Connacht ahead.
After eight minutes Parks slotted home his second penalty after a good spell of pressure that saw the hosts pushing towards the Dragons try line before the infringement.
The visitors were given the opportunity to halve the deficit in the 13th minute after a high tackle, but Lewis Robling's kick sailed wide for the Dragons.
Connacht continued to press, Fetu'u Vainikolo going on a surging 60m run, and although he found Rob Henshaw, some last-ditch defending stopped him inches from the try line.
With 19 minutes on the clock though, the hosts were awarded a penalty try after persistent pressure, with Parks adding the extras for a 13-0 lead.
Connacht maintained their dominance, centre Dave McSharry breaking through the Welsh region's line and his pass found New Zealander George Naoupu who crashed over.
Parks made it four from four when completing the try to give Eric Elwood's men a 20-0 lead, but Darren Edwards' men registered their first score from Tom Prydie's boot two minutes before the interval.
Within one minute the visitors were awarded another penalty, but Robling again failed to convert, Connacht going in 20-3 ahead at the break.
Two minutes after the restart Prydie converted his second penalty of the evening after the hosts were penalised for not releasing the ball, but the Dragons remained well behind.
Elwood's troops registered their third try of the match in the 44th minute as McSharry again proved instrumental, his pass finding Danie Poolman who went over, Parks converting to make it 27-6.
On the hour mark, Parks kicked a perfectly-judged drop goal from a tight angle to extend the hosts' advantage and maintain his 100 per cent record for the evening.
In the dying stages though, replacement scrum-half Jonathan Evans went over to score the visitors' first try of the evening, although Prydie missed the conversion.
The win is Connacht's first in five RaboDirect PRO12 matches, and consigns the Dragons to their fourth consecutive league defeat.
|Connacht Rugby Score Card|
|Newport Gwent Dragons Score Card|