OP, the adjudicator cannot exercise discretion, this lies with the council. However, they may refer a refused appeal back to the authority with a request to reconsider the matter.
PCN - early morning pick-up.
Where is your reply to their email requesting that you or your child's father accompany them on the trip? As I understand it, your argument would be that you expected the school to put your car on their exempt list and notify the council accordingly.
PCN - Parents' evening. By your own admission you simply forgot the time. Discretion is all you've got IMO.
For both, the fact that you were entitled to an exemption in any event. But IMO you need to accept that this goes to mitigation only and is not a defence, this would be that you held one at the time, but you didn't. But this would be a powerful reason for the adjudicator to refer the matter back to the authority with a recommendation to cancel. NB. councils have limited time in which to respond- 35 days I think- failing which they're deemed to have accepted the recommendation and the PCN(s) would be cancelled.