Holiday leave is an entitlement (to be taken within the holiday year) and refusing leave at a particular time for good reason is unlikely to compromise your employees rights. Be careful, however to avoid any accidental discriminatory component in your decision making, e.g. being more flexible for one sex than for another.
Employees also have some limited rights to seek time off in the school holidays, your holiday rota notwithstanding.
The right to take parental leave applies to all parents of children under 5 (even if they have other children over 5). Employees can take up to 18 weeks per child provided it is taken before the child’s fifth birthday. Employers need to have sound reasons for refusal. On the other hand such leave need not be paid and it has to be taken in blocks of one week.
Parents (and others with dependents) can take emergency time off. Again this is unpaid, but it can be taken without notice provided there is a legitimate reason; for example if a child is ill and cannot go to a holiday club. However keep in mind that it is intended for emergencies only. You will have to tolerate a day or two to allow for emergency arrangements to be made, maybe you might need to allow a week, but certainly not the entire holiday.