This worries me. Do airports have enough parking spots if all planes were to be grounded at the same time?

There's enough space, but not enough gates. If they think the stop may end in a reasonable time some planes will be pulling away from their gates even without a departure time to make room for the landing planes (or rather, they will have missed their departure time and not have a new one). Landing planes will be able to disembark, and probably won't be boarded again until it's resolved and enough departing traffic has cleared that it's obvious that those flights won't be canceled for today as well.

The boarded and grounded planes may start disembarking passengers at some point, especially if their crews are hitting their maximum time before mandatory rest. These delays will probably prevent replacement crews from arriving.