Their children are not even allowed to sit exams for free, which they are trying to get changed.

No-one sits state exams for free. Schools are charged. It is part of their budget.
Parents just don't ever see it as an item. Fees at an FE college include the exam fee.

As an FE college and examination centre the college where I worked often had people who had not studied in the college taking exams.

Some of them were home-educated.