The Bolton-born hitman fired in a 71st minute equaliser to cancel out Paul Bolland's 10th minute opener as the Shakers clinched a deserved point.
But Chris Casper's team should have come away with all three points after dominating the second half.
"The goal was fantastic, I loved it, hopefully it's one of many," said Parrish. "My dad will keep reminding everyone in our house of the game.
"You can't believe we were the same team in the second half. We got a good talking to at half-time and realised we'd got to go out and prove something.
"Keith Alexander has told me to get forward and hopefully contribute to the goals.
"The gaffer has always been putting me in there anyway, but with the height of Ben Futcher, they have kept me back at the moment.
"There is only Paul Morgan at the back for corners. The last thing you want is them breaking off a corner when all the lads are forward."
Substitute Andy Bishop played a key role as Bury gained second-half superiority. The on-song striker, who underwent a painkilling injection for a knee problem in midweek, replaced new boy James Dean at half-time.
And while he could not produce the goals, as he had did in last week's victory at MK Dons, he asked plenty of questions of the Mariners' defence.
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