Just 14 points separate the Griff and Coton-based side and the relegation zone with six games remaining in Midland 3 West South.
Park should have enough points in the bag to ensure survival but an eighth defeat in nine games has left them looking nervously over their shoulder.
It all started brightly on Upton's heavy pitch which hadn't seen action for 13 weeks due to heavy flooding.
Hooker Gary Lakin crossed the line and Lewis Brightmore registered five points with the boot to cancel out the host's first try.
However, tries either side of half time put Upton in control before Aaron Willis gave Park a fighting chance with a converted try, that Brightmore then added to with a penalty.
Upton extended their lead with a rolling maul, before Park bullied the host's scrum over the line allowing Willis to touch down for his second of the afternoon.
Park are back in action on Wednesday evening when they take on Southam in the quarter final of the Warwickshire Shield, in a game being played at Nuneaton Old Edwardians' Weddington Road ground, kick off 7.30pm.