Wednesday started strongly and were gifted a goal when Tigers debutant Danny Mills, on loan from Manchester City, was harshly ruled to have handled during a challenge on Burton, who converted from the spot.
Parkin levelled the scores on 11 minutes after Craig Fagan cleverly lofted the ball into the area for the big striker to head beyond the hapless Brad Jones.
Hull continued to push forward and managed move ahead after 17 minutes as Andy Dawson volleyed the ball into the area and Parkin was there again, this time superbly volleying the ball home.
Wednesday tried to find the equaliser from the start of the second half and came close through Glenn Whelan, whose shot from 20 yards took two deflections on its way just over the bar.
Hull went close to extending their lead with a swift break after Lloyd Sam lost possession on the edge of the home penalty area. Michael Bridges took the ball 40 yards upfield before picking out Parkin, but he could only fire a shot over the bar from just outside the area.
Midway through the second half Marcus Tudgay saw a powerful left-foot volley from 20 yards palmed away by Boaz Myhill.
Soon after Owls substitute Chris Brunt found himself in space near the Hull box and struck a right-footed effort from 25 yards, but Myhill was again able to palm the ball to safety.
Wednesday applied some late pressure and Myhill again denied Tudgay and the Tigers held on for the victory.