OP, you knew that schools are no longer allowed to give children time off for a simple family holiday.
You knew that if you took your children out for a simple family holiday then that would be recorded as an unauthorized absence.
You knew that an unauthorized absence could well result in a fine of 60 per parent per session per child.
You have been fined. That is not unexpected. You are lucky that your school/LEA has taken the four days as one session and not as four.
These fines are not fair, IME. Parents who can't otherwise afford a holiday-time holiday (be that a week in Skegness or a fortnight in the Seychelles) are being denied a holiday. Parents who cannot get holiday-time off work are being denied a holiday. There are other reasons why a holiday in term time would be better for parents, but is denied.
The point is, the class teacher cannot authorised this. The HT cannot authorise this.
Teachers' training days do not affect holidays. Schools should be open for a set amount of days in the year. Your child should attend school on those days.
Teachers do not strike because they want a day off. They strike because they want to be treated fairly. Because they work very hard and provide an essential service and deserve to be treated accordingly.
You could try fining the school for absence during a teachers' strike, but as it is not the school fining you for your absence during a holiday it would be futile.
You could try fining the LEA for absence during a teachers' strike but as the LEA isn't authorizing the strike, and as the teachers are, in effect, being fined themselves as they do not receive that day's pay, it would be futile.
Or you could accept that you have been given a fine, with good reason, and simply pay up.