I understand what you are saying, but if you play a course and it has the par stated on the card that doesn't match those numbers do you change what the card says? Do you go into the clubhouse afterwords and tell them it's wrong? I'm not arguing that the hole is crazy short for a par 5, that much is obvious. If I were to talk about getting an eagle on this hole I'd qualify it by saying "but it was only 360 yards so it was basically a birdie on a par 4 anywhere else". The guy was just asking if it should count as an eagle, which I do just for the fact that the course said it is.
While that is possible I think it is more likely the tees were only rated for women and if they had both women and men ratings the par would appear on the scorecard as 5/4, at least that is fairly common here when forward tees on holes this short have both ratings.
In any case regarding distances (my emphasis added):
If the card says its a par 5 then its is a par 5. Some courses have a different par for the short tees.
For example, hole number 9 on a course I play, it plays as a par 5 for the shorter tees. I understand if they happen to rotate forward, but the par is the par for the hole. They just expect those will less skill to play forward so they adjust the par accordingly. If the hole stays a par 5, then it plays as a par 5 for all the tees. That is how the course is set up. Doesn't matter if it is 350 yards or not, its a par 5.