McLean's 35-yard pile driver six minutes from the interval was fit on its own to win any game.
However, Parkin clinched the points in the 62nd minute when he controlled a Dougie Imrie cross before guiding the ball past Dees' veteran keeper Rab Douglas.
Dundee had their moments with midfielder Ryan Conroy hitting the post just before the break and Mark Stewart and John Baird missing great chances but it looks like being a difficult season for Barry Smith's team.
Both sides made changes, former Aberdeen defender Davide Grassi made his Dundee debut and Stephen O'Donnell also started with Neil McGregor and Kevin McBride dropping out.
Saints striker Stevie Thompson returned from suspension to replace Lewis Guy, banned after being sent off against Inverness last week, while Graham Carey and Jon Robertson came in for Gary Teale and David Barron, who both dropped to the bench.
The second week of the season had been designated "sell-out Saturday" and although that failed to materialise there was a decent and enthusiastic crowd, most of whom were there to see the SPL new boys play their first home game of the campaign.
The early stages, though, were a bit crash-bang-wallop with neither side able to take control.
In the fifth minute Douglas made a decent save from McLean after the Saints midfielder had taken possession of a short Carey free-kick 25 yards from goal.
Dundee midfielder Conroy then tried his luck with a long-distance drive in the 12th minute but the ball sailed wide of Craig Samson's right-hand post.
At the other end, Thompson flashed a volley over the bar from the edge of the box and then midway through the tepid first half the former Rangers and Dundee United striker headed a David Van Zanten cross over the bar.
The game was going nowhere fast when McLean picked up the ball 35 yards out before arrowing a drive past the flailing Douglas.
It was a goal out of kilter with the football being played but one which thrilled the travelling support behind the opposite goal.
The Buddies fans, though, held their breath seconds before the break when Conroy's drive from 25 yards beat Samson only to rebound wide off the post.
The second half began at a frantic pace but with little attention paid to keeping possession.
Thompson was booked in the 51st minute for clattering Dens Park midfielder Iain Davidson and the home fans wanted him off two minutes later for a challenge on defender Kyle Benedictus but referee Crawford Allan gave the former Tangerines striker the benefit of the doubt.
Samson was the Buddies hero in the 55th minute when he made a point-blank save from Dundee striker Stewart who had rather easily shrugged off the attentions of Paisley defender Darren McGregor before getting his shot away.
Moments later, Conroy's deep corner from the left arrived at the feet of Davidson 14 yards off but he snatched at the chance and cleared the bar.
Dundee had a penalty claim turned down just after the hour mark when Stewart went to ground in the box following a tackle by the backtracking McLean, their frustration compounded a minute later when they found themselves two goals down.
Imrie's cross from wide on the left was controlled by Parkin and he had all the time he needed to control the ball and place it low past Douglas for his first St Mirren goal.
The home side had a great chance to get back into the game in the 65th minute when Jim McAlister stood his cross up at the back post for Stewart but he mistimed his jump and headed over the bar.
The miss punctured the hopes of the home fans who had more to moan about in the 74th minute when Davidson had the ball in the net only to see the flag up for offside.
There was no doubting Dundee's work rate and determination but they looked a bit short of quality, shown when Baird missed a sitter in the final seconds of added time when he lobbed the ball over the bar with only Samson to beat.