I can quite unsderstand that your parents are upset and this is an error that many people make in all innocence.
It is the responsibility of the person making the booking to ensure that all names are correct when they provide them to the airline. These names should match the details in the passport for each passenger travelling.
Each airline has their own policy on 'changing' a name with most being very strict and insisting that a fee is paid to change the name as soon as the booking is confirmed. Others such as easyJet give a days grace before charging and British Airwasy also do have some leeway.
Your first action should be to call their helpline and see if they will waive the fee, however I fear that they will indeed impose this charges as most airlines will do. Explain that you tried to call as soon as you made the booking. They may also ask for any increase in the fare between that paid and what the fare is on the day you make the name change.
This may well have been a costly mistake, so let's hope Monarch show some leniency as a New Years resolution! Do let us know how you got on.
Travel expert, www.travelsupermarket.com