Flight has been evolved independently 4 times, gliding much more - how does that square with the principle of parsimony? (ie, if we hadn’t discovered pterosaurs yet, we could make the argument that flight evolved 3 times and probably not more due to the principle of parsimony, but we would be wrong)

the principle of parsimony is just that, given equal evidence one way or the other, the simpler explanation (flight only evolved 1 time in birds) is more likely then the more complex one (birds gained flight, lost it, then gained it again).

The fact that there are only 4 times it's happened that we know about is just evidence that it's a fairly rare thing to evolve.

But given that it evolved once in birds, after an event that wiped out all of the flying animals, wouldn't the principle of parsimony imply that birds have the lowest barrier to gaining it again to take advantage of that now empty niche?

that depends on how you think loss of function and gain of function works at the phenotype level. For example, I choose to interpret these sorts of situations in the way you do: birds that lost flight still had 99% of the flight capability, being maintained under functional selection for other uses, and so only a small amount of change would be required to regain the ability.