Just a thought - if you have a few pupils who's parents say that the time allocated to them for their exam is unsuitable, who's responsibility is it to sort out?

If it's a genuinely serious problem then as a teacher I'd make the effort to sort it out. It probably should be the applicant who does this anyway.
I do make sure to point out to parents that there really isn't any control over time and date before the entry goes in. I also tell them that it's possible that they will have as little as 2 weeks notice. I generally take it that accepting the entry going in indicates that they are aware of this. There are sometimes genuine exceptions.

As I enter the pupils, I take responsibility for trying to change the exam time, even if it means that they have to go to another centre. So far, I haven't had a problem and the board are always very helpful.

I always warn the parents in advance that the exam session is very likely to be on a school day between the hours of 9 - 5. This way they know that they will either have to take time off work to chaperone their child to the exam, or that they will have to find someone else to take the child along. Very rarely have we had to change the times. I did have one pupil ask if I could chaperone the pupils to the exams but as it's usually at a time when I am teaching piano, plus I don't drive, I said that I can't offer this service.

I have occasionally had to make changes, when exams have clashed with residential school trips. The Board have always been very helpful. Last time this happened, the trip involved six children so we were moved to another centre. Years before, I managed to get an exam changed to an earlier date for a child.

I have occasionally done this when something really significant has made the appointment "unsuitable" - for instance, when the inquest on the suicide of someone's son's father was scheduled on the same day. In that instance the Board were very helpful. It is more difficult to change days or centres for one candidate, because the accompanist has to be available too, and obviously has to charge that person extra for a one off - especially as it is a round trip of 50+ miles for one 15 minute exam.
Nowadays, as I have home visits, everyone knows what day the exam is at the time of entry (a month in advance), just not the time, so there is really no excuse for it to be "unsuitable", unless for extreme circumstances. As regards the time at centres, we are allowed to change appointments around within our allocated time slot.