I am a teacher and so have some idea from the other side.
In a school with 360+ pupils it is very difficult to know all the circumstances of everyone and unfortunately, parentas dont always explain situations to us, either because of privacy or they think it isnt relevant! Class teachers are usually aware of specific situaions if we have been informed but we certainly couldnt expect the office staff to know about everyone.
Sadly, if both paerents have parental responsibilty, there is very little we can do with regards to picking the child up from school etc. If an absent parent turned up at the end of the day, we would not be physically allowed to stop them. I can completely empathise that this is a difficult situation but we just don't have those powers. If there was some sort of order in place then obviously the situation is different. Also, if a parent turned up during the middle of the day, I am sure common sense would prevail and some sort of question would be raised as you aren't allowed to take children out of school without warning!
With regard to holidays ( sorry to be boring but I dont condone them in school time anyway as the children miss so much and it is disruptive but I completely understand the cost issue as I always have to pay the school hols premium!!) yes the school can give permission but that's for the absence from school at their discretion its not necessarily permission for the parent to take the child from the resident parent, does that make sense?! The father still needs permission from the other parent surely!? I googled about permission from absent parents about goign on holiday and techincally, if he doesnt have your permission, he could be accused of abduction if he just took the child!?