Park visited their local rivals, a vastly improved side, taking a decimated squad.
The game was relatively even until a cross caused problems at the back post and the hosts tapped in.
Park came close with a diving header from Carl Longmore that was well saved and then a Danny Stewart shot forced a brilliant save.
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The stuffing was knocked out of Park when a dubious penalty was awarded following a Rob Warner challenge on the edge of the box.
Phoenix tucked it away neatly for a half-time 2-0 lead.
The second half was almost a non-event, but Phoenix scored in the 72nd and 75th minutes, both from close range.
The close season at Kiveton will see some ground developments as a result of winning a Sport for All grant.
New security fencing and CCTV will be installed together with ball stop netting and a new garage.
Kiveton Park will host their Junior Football Tournament on Sunday 2nd June, with 40 teams taking part.
Hot food and drink will be available and activities will include an inflatable penalty shoot out game.