There was a try apiece in a free-flowing first quarter, Kahn Fotuali'I striking first for the Ospreys before Willie Faloon replied with his maiden effort in Connacht colours.
Parks slotted two of his three penalty attempts to move Eric Elwood's men 13-7 ahead for half-time.
Penalties cost the westerners when going down to the Scarlets last weekend, however this time they gave the Ospreys - conquerors of Ulster in the previous round - precious few chances to close the gap.
Matthew Morgan slotted a 69th minute penalty, but Parks rubberstamped the home victory with two second half penalties and a late drop goal.
Although the visitors had the better of possession early on in Galway, Connacht snuck into a seventh minute lead as Parks punished a scrum infringement by Cai Griffiths.
Playing into the wind, a good spell of ball retention duly paid off for the Ospreys on the quarter hour mark. Morgan was held up short before his half-back partner Fotuali'i quickly tapped a penalty and burrowed over from close range.
Morgan converted but the Ospreys' lead lasted barely four minutes as the hosts hit back in determined fashion.
Connacht attacked from a solid scrum with John Muldoon and Michael Swift linking on the right wing, the ball was recycled and flanker Faloon's clever line took him over past two defenders.
Parks landed the difficult conversion for a 10-7 lead and was narrowly wide with a subsequent penalty from distance, with the Ospreys front row coming under further scrum pressure.
Faloon was unceremoniously dumped in an apparent spear tackle that the match officials failed to penalise, as the game continued to canter along at a brisk pace.
Connacht had the final scoring opportunity before the break and Parks stepped up to widen the gap to six points, rewarding his forwards for another power-packed scrum.
Brett Wilkinson and captain Nathan White had the beating of their opposite numbers, with young number 8 Eoin McKeon, who got through a huge amount of work around the pitch, impressing once more for his home province.
The wind advantage was with the Ospreys for the second period, but a Kyle Tonetti block and a subsequent three-pointer from Parks kept Connacht on course.
The Ospreys failed to follow up on a promising lineout maul, while Parks' cross-field kick helped the returning Gavin Duffy bring Connacht downfield.
The former Scotland international drew a drop goal attempt wide and as time ticked by, the Ospreys continued to make errors and struggle for accuracy.
Alun Wyn Jones' first start for the Welsh region since October was not going to plan, and Danie Poolman went close to collecting a loose ball during a dangerous counter attack for the hosts.
Minus a raft of Wales internationals, the title holders brought on Rhys Webb and Jonathan Thomas and a gradual improvement led to Morgan's lone penalty goal.
Steve Tandy's charges probed from the restart, yet a crucial turnover from Duffy relieved the pressure on the men in green.
Then a pinpoint up and under from Parks allowed Robbie Henshaw to hammer Richard Fussell in the tackle and the Ospreys full-back coughed up a penalty for holding on.
Parks clipped the resulting place-kick over from the left for a 19-10 scoreline and the final nail in the Ospreys' coffin was driven in by the out-half's sweetly-struck drop goal, two minutes from time.
Those kicks took the losing bonus point away from the Ospreys and James King's yellow card for a dangerous tackle on Faloon saw them finish the match with 14 men.
|Connacht Rugby Score Card|
|Ospreys Score Card|