It was one of only two games played in the NPL on Saturday, due to a combination of poor weather and the involvement of several teams in the first round of the FA Trophy.
A Chinedu Osadebe header shortly after the hour mark proved to be the difference at Horsfall Stadium as the hosts picked up just their third league win of the season.
Elsewhere, Warrington Rylands missed the chance to close the gap to league leaders Radcliffe after being held to a goalless draw at Lancaster City.
Nevertheless, Saturday's point saw the Blues stretch their unbeaten league run to 13 games and maintain their unbeaten start to the season on the road.
The afternoon's other scheduled match between Whitby Town and Gainsborough Trinity was postponed due to a waterlogged pitch at the Towbar Express Stadium.
Results - Saturday 28 October
Bradford Park Avenue 1-0 Marske United
Lancaster City 0-0 Warrington Rylands
Whitby Town P-P Gainsborough Trinity
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