Has it ever happened where an arriving plane had to take off immediately because there was no parking space at airport?

Is it mandatory that all arriving planes have a parking space available?

What IF it were to happen? What would be the solution?

If by parking space you mean a gate to pull up, then yes that happens all the time. I've personally been on planes which landed with no gate available. Airports have various holding areas where planes can sit away from the gates.

If there's no holding area space than the plane simply doesn't land. Air traffic control requests they circle until space is available, that has happened on flights I've been on too.

No - there are not even designated "parking spaces" at most airports. Planes can be stowed on the ramp where ever they fit - if you look inside any shared hangar, the planes are often packed in any way that allows them to close the door.
Airports have a lot of open space - if there's not room on the ramp they will find an out of the way location else where.

They wait.
This is quite common when a plane arrives at their scheduled gate and the previous flight is running late.
If it's expected to be a short wait, the plane waits in a designated waiting or holding area on the apron.
If it's going to be awhile for the previous flight, the waiting plane will likely be directed to another gate that's open.
There's really no reason that would cause the plane to takeoff again unless there was like a war

Nope. They wouldn't let the plane land if there wasn't somewhere for it to go.

If a plane had landed before a gate was available, they'd taxi around on the ground generally rather than take off again.

NO
If there's no place for the plane to fit on the ground the plane does not land.

Is it mandatory that all arriving planes have a parking space available? NO they just need a clear runway to land and for the next flight to land behind them enough space to get off the runway.

In emergency situations planes can be hauled off to the grass to make room for a plane to land.

Planes can be towed very close to each other and stacked up on available space.
It happened during the first hours when the skies were cleared of all flights in September 2001.

That there's no gate available the plane would not have landed in most situations.
It does happen a diverted plane may just land and wait at some other Airport in a parking/waiting area" and then take off when their destination Airport becomes available.
A decision made at the moment by ATC and Airline operation centers.

A picture of mass parking in an emergency situation

SIXTY just parked planes are in the first picture.

At a time when the world, for many, seemed to be falling apart, many Canadians, in Gander, Halifax and many other places, came together to open their houses, stores and hearts to perfect strangers. And I dare to predict that neither the Canadians nor the strangers will ever forget the experience.
Bruce Ricketts

They would never just take off again. By the time they arrive at the destination airport, their fuel supply will be very limited and would probably be below the FAA rules about needed reserves to fly to another airport.

Planes can sometimes wait for long periods of time on unused locations of taxiways, aprons or gate areas.

If they don't have an available gate, then they just wait out on the tarmac somewhere until one becomes available. If there's no room anywhere (which would be rare), then the plane would be diverted. In any case, it would not take off after landing.

No, and a plane would never turn around if it hadn't landed back to it's original airport as you asked in one of your comments. There are fuel issues at minimum there. There's always room for planes to park. If they don't have a gate they wait for one to open up. It happens from time to time. If the plane has landed they wait. In flight, if for some reason they know of an issue at the airport, it's possible the plane would be diverted to a nearby airport. It would be very rare, but it has happened.

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