If wondering, RWD vans can also get stuck on wet grass as well. Especially if the PU / TT has been sitting on softer grass and its tires have sunk in. Even sinking in 1/2" can make a RWD vehicle spin their tires. And, instantly get stuck. Been there several times - even with RWD vehicles.
For your ex-PU (now upgraded to small TT), I would recommend parking on hard boards. During storage, I hear its best to park on wood. And if possible, put large patio stone under the wood boards as well. Thus, making its wheel contact area even more firm.
To view the trailer pad for my previous 19ft TT, surf:
Even with a parking pad, I had to pick optiminal times when to back our TT across our home's front lawn. Too soft and the TT tires sink in (cutting the sod like a knife). When firm but grass was too wet (re: during morning dew), my Safari van's Rear Wheels would spin.
At my seasonal camp site, I sometimes pull a boat up the cement ramp. A steep ramp but being on hard cement, my FWD mini-van does it (without spinning). But, if my front wheels get on soft ground or wet grass, they loose traction. Thus, I pick my time of day to drive across wet grass. It's true that AWD & 4x4s have little to no problems driving across wet grass. But with mini-FWD vans, its unfortunately that one must pick the ground conditions... Or, park the PU/TT on wide boards - to get some initial movement in your first towing pull...
For mini-van and TT, I'd recommend a WDS (Weight Distribution System) like Reese brand as well. If possible, get a WDS with dual sway control as well. Especially for short Wheel Based Tow Vehicles - like mini-vans and future Ford Explorer vehicles.
Hope this helps....