Parrish scored his first goal in more than 140 first-team appearances as Jim Bentley's side came from behind to record their first win in seven games.
The Shrimps got off to the worst possible start when they went a goal down after just six minutes. Matthew Saunders swung in a corner from the right which flew across the Morecambe penalty area before Brian Woodall produced a fine finish to volley past Barry Roche from 12 yards.
It was a goal that silenced the home crowd and the Daggers almost doubled their advantage after 15 minutes when Stewart Drummond headed towards his own goal forcing a vital clearance from Kevin Ellison.
Morecambe raised their game in the second half and levelled after 53 minutes with Jordan Burrow scoring his first goal of the season. Lewis Alessandra crossed to the far post and Ellison headed back into the danger area with Burrow in the right place at the right time to bundle the ball past Chris Lewington from close range.
Morecambe continued to attack the visitors and were denied a second with an excellent save from Lewington to tip away a sweetly-struck Ellison strike from the edge of the box.
Gary McDonald then saw a close-range volley superbly cleared off the line by Scott Doe before the Shrimps took the lead with 11 minutes to go when Parrish steamed in at the far post to force the ball over the line from close range after a superb run and cross from Alessandra.