I understand the Pauli exclusion principle like this:

1. For two electrons to occupy the same state their spins must be opposite.
2. If the two electrons are in different states (different spatial wave-functions) their spins are allowed to be parallel if this is otherwise energetically favorable.

The question is this: How different must their wave-functions be for them to allow parallel spins?

Or posing the question differently: If we start to "deform" one of the wave-functions so that it becomes increasingly similar to the other wave-function, at which point will the spins be forced to become anti-parallel? At the point where the wave-functions becomes identical?

I hope this makes sense.. Anyway, thanks in advance!