My model was produced 12 years ago and there have been no reports that I can remember of this problem. Just to be sure, I've just tested it in FSX and P3Dv5 and it is stable, both when parked and when moving. This is P3D:
As Steve says the instability can be caused with the static_pitch and/or static_cg_height being out of sync with the contact points, but it can also occur when the CG has been moved.Try my original aircraft.cfg file (attached) to see if that solves the problem.
John