Unfortunately not all schools are aware of, or correctly informed about, SPL. This page seeks to provide some advice should you find yourself in this situation. Over the last two years I have spoken to many women who have faced some push back with their applications - please take courage from the fact that every single one went on to get the leave that they had requested and were legally entitled to. It can be helpful to refer them to this site when you first mention SPL (many schools also turn to ACAS for advice).
https://www.acas.org.uk/shared-parental-leave-and-pay/taking-shared-parental-leave - this page specifically states that 3 blocks of continuous leave applied for individually cannot be refused.
Employers cannot refuse applications for three blocks of SPL *IF* each block is requested on its own separate form (government form 2, or your schools version of it). There is only one legal ground for the refusal of SPL if requested in up to three blocks - which is that you don't meet the general eligibility criteria for SPL.
Your school can ask you to attend a meeting to discuss your application, within the 15 day window that they have to reply to your requests for leave. These can be really useful. I advise the following; you don't have to go alone, you can ask to take your partner, a work colleague etc, you equally don't have to attend full stop - your application can not be legally refused and if you read my story with my first child I declined the offer to meet saying that I didn't need the additional stress in pregnancy, and finally remember that you don't have to agree to anything in the meeting. I would advise that you thank them for their thoughts and feedback and say that you will now go away and discuss it with your partner - this avoids the pressure of having to tell them your decision in the moment.
If your employer does (mistakenly) refuse your leave (and for many it won't get to this point as they will seek advice before formally responding to you and realise you are in the right) then I would send the following letter in your shoes. Please feel free to use this/ adapt it to your circumstances etc.
Thank you for your response to my application for Shared Parental Leave.
My application was made after taking the time to research eligibility for SPL and how the periods of leave work. I have considered the needs of the school, but also the needs of my family during this period of maternity leave. SPL is becoming increasingly widespread amongst teachers as it essentially levels the playing field with the many other professions who accrue paid holiday that can be built into maternity leave in a way that teachers can't.
Both the gov.uk and ACAS websites are clear that each parent can make three applications for separate blocks of SPL and that if each application is made separately (so the leave is deemed to be 3 blocks of continuous leave rather than discontinuous leave) then the only legal ground for refusing leave is if the parent doesn't meet the eligibility grounds in the first place.
I therefore ask that your decision be reviewed as I believe it to be unlawful, even if not intended to be so. The law is clear that if a refusal is not based on eligibility then the refusal itself is invalid and the leave must be granted as requested. I would be grateful for a response in writing.