Unfortunately, their video shows, (0.03) that before your father entered the YBJ, the taxi had already moved forward to turn left. Your father therefore moved into the box without the space to exit, which the taxi had taken. Basically, he should have waited outside the box until the taxi had completed its manoevre. This would then have shown his exit was blocked. Yes, I know decisions have to be made in split-seconds, but that's motoring, I'm afraid.
This may not be the end of the story, because there could be a 'technical' appeal based on council mismanagement of the enforcement process, so wait a bit to see if any such appeal argument is suggested. However, don't miss the deadlines stated in the PCN, (discount period, and period to pay or appeal)