That does seem odd! First thoughts are bad oil pressure sending unit, or a loose or faulty connection. Pressure sending unit lives on top of the passenger side bank and has 2 spade connectors. I'd double check and see if those are tight, if they are then I may go ahead and replace the sending unit. Could be the gauge but after sitting for a while combined with the age of it, my money is on the sending unit.
Either way not necessarily something you want to chance so you may want to see if there is a knowledgable shop nearby, or if you can hook up an analog oil pressure gauge to verify what you are seeing, as you don't want to keep running it if it is actually dropping pressure.
At the end of the day I doubt the oil pressure is actually disappearing, rather false alarms from the electronics as these motor are not known to have oil pumps fail...
Another thing you could do is pull the oil pan and check the oil pickup tube but if it was clogging up I doubt it would go straight to zero. Just typing out loud now
Good luck, let us know how it goes, and welcome to the forums!