@ZiriForGood
Another hilarious idea is planning only for days this employee is there. That would effectively limit number of part time workers in a department to one or two.
Well, it is pretty clear from what the OP has posted that she has no experience of managing PT staff, so clearly, in this case that isn't an issue.
However, where I work, there are a lot of PT staff, and the agreement is, every part timer works Wednesday. Therefore it is just a given that any training or meetings that need everyone there, get booked on Wednesdays. Nobody has to change their days or work extra.
Of course in the case of the OP, she had a choice of 4 days to book it.
In this case the employee was allowed part time work with mutual understanding that there will be some switching to meet business needs and education might be a valid case
No, she said herself it was for project delivery . That is completely different from saying it is for "anytime her incompetent managers fancy telling her they want her to come in"
One issue might be communication. Maybe it wasn't clear that you consider that event as the important one to use up one of the swaps.
Except, the OP has clearly stated
a) it isn't critical that the PT employee attends
and
b) The 4 day week arrangement was agreed, with the caveat the PT employee comes in occasionally on her NWD for project delivery