Why fight over the storage unit?
Drop the subject.
In June, go over there when you have time, alone. Take one box for shredding, one box for keeping, and one box to take home and decide about later. Store the keeping box on site, but mark it so you don't re-sort it.
Next trip, take three more boxes.
*If* your folks ever get the notion to go over and look through the unit, will they even notice that some of the junk is missing?
When you're down to just the keeper boxes, decide whether to get a smaller/cheaper unit or move it to their place. Don't ask. Don't discuss. Just take care of it for them.