The visitors took the lead in the 25th minute when Junior Brown's right-wing cross to the far post found the unmarked Parkin and he volleyed home.
Stanley could have equalised immediately after as James Beattie played in a superb cross but Francis Jeffers just could not connect from four yards out.
Fleetwood, who had gone five games without a win which had seen them drop out of the top seven, were denied a second in the first half when goalkeeper Paul Rachubka kept out David Ball's goalbound effort with his legs.
However the Cod Army doubled their lead after 62 minutes when burly striker Parkin was put clean through by Ball and, with only Rachubka to beat, stroked the ball home.
Number three came when Parkin flicked Shaun Beeley's cross into the back of the net at the near post after 70 minutes, while Jean-Michel Fontaine hit the crossbar in injury time.
Stanley's Rommy Boco and James Gray forced late saves out of Fleetwood 'keeper Scott Davies but the Reds have now not scored for three games.