Any hazards that exist while the motorhome is moving are also present while the vehicle is parked. It's no more, or less, dangerous to operate a propane refrigerator while moving than while stationary.
Propane appliances should be turned off near gas pumps, due to the remote possibility that a spark from the DSI (Direct Spark Ignition) could ignite gasoline vapors. That hazard doesn't exist with diesel fuel.
For what it's worth, I've responded to a grand total of two RV fires in my 38 years as a firefighter - both were parked at the time.