Can all airborne aircraft be stored at their origin or destination airport?
Short answer: Yes.  

You need to distinguish between short-haul and long-haul fleets.  

For example, when it is 4 a.m. in Central Europe, almost all the aircraft which fly short-haul routes throughout Europe during the day will be on the ground somewhere in Europe simultaneously and there must, therefore, be ample parking for all those aircraft (though not necessarily at gates). Similarly for other parts of the world: when it is 4 a.m. in the central U.S., all the short-haulers will be parked-up somewhere in the U.S.; when it is 4 a.m. in South East Asia, all the short-haulers will be parked-up somewhere in South East Asia, and so on. There must, therefore, be ample parking for all short-haulers globally.

For long-haul fleets, it is a different matter. Such aircraft spend up to 16 hours a day in the air, and therefore well over half the aircraft will usually be airborne at any one time, round the clock. It is highly abnormal to have all long-haulers on the ground simultaneously. If all those long-haulers need to suddenly be grounded, it can cause serious problems, as happened after 9/11 when the entire transatlantic and transpacific fleets suddenly stopped flying. However, even then, the airport operators did manage to find sufficient parking.