I think your responses here are rather strange OP. You wanted advice about summer pay, so posters have given you helpful advice on how to achieve the best results in your business, but you haven't responded with any indication that you understand that you have to have a business plan and terms and conditions in place in order to succeed. Instead you came back to the thread to comment on one pupil's desire to achieve high grades, exams and give other irrelevant information. 🤷♀️
Your posts show a real lack of awareness of how to run a business. The situation you describe is more like what you'd expect with a casual babysitting job - instead of you being in control, it seems you're just coasting along waiting to see whether parents will want you around for holidays, and tutoring in general. Plus, they just drop you at short notice because you didn't discuss terms and conditions initially. Not what you'd expect from a professional tutor who needs to earn a living from tutoring.
You have to work out what yearly income you need to earn and then work out how many students you need and the price you need to charge to achieve this. Factor in tax, pension, business expenses etc when working out what income you need.
Parents should be made aware at the very start of price, payment dates, what happens during holidays, what happens if they need to cancel (e.g. no refund), that a month's notice is required to end tutoring, whether a deposit is required to secure tutoring after the summer holidays, etc, etc.
As others have mentioned, most parents don't want to pay for holidays so that's why you charge a higher price to account for this and save that extra money to use during the holidays, when you don't have work. It's up to you to budget effectively so that you have this extra money to live on over the summer and other holidays such as Easter and Christmas. Students who sign up for all year tutoring should be charged a discounted rate.
With your current clients, you can tell them you're reviewing your working model and then update them of the new changes.