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Referee David Rock was at the centre of controversy as Welling's relegation woes deepened in Vanarama Conference on Saturday.
Trailing to Ollie Palmer's ninth-minute goal, the Wings appeared to have been handed a route back into the game when new signing Tarshan Adeyinka was hauled down by Aristoe Nsiala.
The assistant referee flagged for the spot-kick - which would have also seen Nsiala red-carded for a professional foul. But Mr Rock, who originally waved play-on, stopped play to talk to his assistant, only to give Grimsby a non-contested drop ball after telling both captains of his decision.
It left a bitter taste for Wings fans as seven minutes later Grimsby added a second goal with the hosts feeling they should have had a free-kick after a foul on Ross Lafayette.
Instead, play continued and Grimsby took full advantage, winning a corner which was headed in by Shaun Pearson.
Despite the scoreline and the lack of goals, Welling could take plenty of heart from this display into their final six games. They need to win at least four to have a realistic chance of staying up.
Ross Lafayette started alongside Adeyinka and the former was inches away from heading in a first-half equaliser from Joe Healy's cross.
At the other end, Jon Henly saved well to den Nathan Arnold and he also tipped over Lanell John-Lewis' header.
Welling were the better side after the break and a goal for the hosts would have changed the complexion of the game.
They nearly got one when Dominic Vose's free-kick hit the top of the bar while Michael Chambers' header was brilliantly tipped over by keeper James McKeown.
Healy drove over and Vose's low shot was saved as Welling kept going but by the end they were left to rue being on the wrong end of the referee's decision.
Welling: Henly, Williams, Osbourne, Chambers, Harris, Healy, Gallagher, Corne (St.Aimie 64mins), Vose, Lafayette, Adeyinka. Subs not used: McEntegart, Fagan, Young, Taylor.
Attendance: 1,042.