I would think since its 1-4x that it would have parallax appropriate for the short ranges that he has tried. Zooming sounds like too many things are moving inside the scope.
A Google search and a visit to Leupold's website has me thinking I have found your issue possibly. There are no vx-1 1-4x20 scopes listed for centerfire rifle use. There is one designed for shotguns though. I don't know what the difference is but me thinks you bought the wrong scope for your intended use. Call leupold and give them the serial number to find out what it is you own.
their sight lists it as shotgun or muzzleloader so I would assume that a muzzle loader would require the same level of accuracy that I've been attempting.
People use shotgun scopes on rifles in order to avoid parallax issues at close range.
The only issue you should have is a vertical point of impact shift at different ranges, but from your description I think you already accounted for that.