Is this a state school, a free school, an academy or an independent fee-paying school?
If it is a state school there is a route to appeal, but as the rest are basically their own kingdoms there's little you can do but pay up.
So if you're dealing with a state school, try the following in this order - don't circumvent the order or you'll create unnecessary opposition.
1. See the head and write to the head personally ie don't be fobbed off with 'in a meeting', 'talk to the secretary'. Schedule a meeting face to face with the headteacher and explain your work position, plus the fact that any work issued by teachers will be done during the week, and any homeworks completed. Don't mention or raise any other child's alleged access to school-time leave - this will give the HT leeway to say you are being prejudiced. And lay it on thick about your leave pattern, stress, etc etc. and the child's good attendance record.
2. Ask the HT to reconsider and to give the decision the next day - don't hold out for a decision on the spot as it will be easier to say no. In the best of cases the HT will need to consult teachers about setting work.
3. If the answer is still no, write to the chair of the governing body with the same request.
If the school your child attends is a 'good' school they are more likely to be a bit precious about this issue. Lairey schools have that many kids unaccounted for anyway that they will probably just be pleased you asked.
Lastly, go on holidy anyway and wait for the bill for the fine to come in. Sometimes these are just empty threats. I wonder if any Abers have direct knowledge of the process used for raising an invoice for a parent - the effort involved is probably counter-productive at school office level.
In lairey schools most families will simply say they've got no money if the school tries to fine them. There is absolutely no prospect of a local authority seding in the bailiffs for £120.