Jonathan Spector's disastrous own-goal meant that the Hammers had got off to the worst possible start and even though Ilan and Radoslav Kovac then gave Gianfranco Zola's side an interval lead, Hugo Rodallega levelled just after the break.
But then Parker struck his second goal of the campaign to leave heartbroken Hull City marooned in 18th spot, six points and 23 goals adrift of the ecstatic East Enders with just two games to play, while Burnley need a miracle too.
The Latics' remarkable late fightback against Arsenal last weekend had given them the luxury of kicking off in 15th spot, fully four points clear of their hosts and having seen his side recover from a two-goal deficit to shoot down the Gunners.
Roberto Martinez made just one switch as Gary Caldwell came in for the benched Steve Gohouri.
Certainly, Wigan quickly carried on from where they left off on Sunday afternoon, taking a fortuitous fourth-minute lead when Spector inadvertently nodded Ben Watson's in-swinging corner beyond his own keeper.
A miserable Monday night on Merseyside had seen West Ham meekly surrender to Liverpool but, following that defeat at Anfield, Zola surprisingly made just one change to his 17th placed side as suspension-free Parker returned in place of substitute Junior Stanislas.
On eight minutes though the Hammers looked all set to level when Ilan played in escaping top-scorer Carlton Cole, who rounded the solitary figure of Chris Kirkland only for the retreating Caldwell to somehow stick out his left ankle and block the shot on the line.
Cole's agony was nothing compared to that of the keeper, who received lengthy treatment for a bloody face injury sustained when he was accidentally clipped by the England striker in that free-flowing move.
The natives amongst the sun-drenched crowd of 33,057 were getting more and more restless by the minute and when James McCarthy - who had earlier been booked for felling Ilan - let fly from 30 yards on the half-hour mark, they had Robert Green to thank for diverting the stinging shot onto the crossbar.
Incredibly, instead of finding themselves two goals down, West Ham were suddenly all-square as Cole beat Mario Melchiot down the left before sending a low cross into the six-yard box where Ilan sneaked between Caldwell and Titus Bramble to steer home his fourth goal of the season.
Then in the fifth minute of first-half stoppage time, Mark Noble's goalbound 25-yard free-kick was acrobatically clawed out from under his right-hand angle by Kirkland, who could only look on helplessly as Kovac beat Caldwell to the looping ball to nod the Hammers into an interval lead.
Within seven minutes of the restart, however, the Latics levelled thanks to another Watson corner that found its way to the far post, where Rodallega bundled the ball over the line amidst the protests of the home defence, whose vociferous claims for handball fell on the deaf ears of referee Alan Wiley.
As the hour-mark approached, Spector was booked for felling Charles N'Zogbia, who dusted himself down before forcing Green to save his consequent 20-yard free-kick at full stretch.
At the other end, having claimed only his second goal of the campaign, the climbing Kovac looped Noble's corner onto the crossbar before Zola introduced Guillermo Franco at the expense of Ilan for the final 20 minutes.
And within five minutes of his arrival, the Mexican substitute nodded into the path of Parker, who virtually guaranteed Premier League football at Upton Park next season when he planted an unstoppable, swerving 25-yarder beyond Kirkland.