There is nothing in law that states that an employee must take all of their paid leave entitlement before they take any unpaid leave in relation to going away on holiday.
Unpaid leave, for reasons of going on holiday, is not covered in legislation.
Sometimes employees have provisions within their contracts of employment dealing with the taking of leave and any unpaid leave that may be granted, so you should first look at their documentation to check this.
In the absence of any contractual provision, how you deal with unpaid leave and whether you allow it is at your discretion, though you should look towards any past instances of a request for unpaid leave in your organisation for guidance in this current situation.
Leave for reasons other than holiday
You should query why the employee wants to take time off in this way. If it is because they want to spend more time with their children, then this leave may fall under the definition of parental leave and can be classed as such, rather than simply unpaid leave.
There would be no difference in terms of pay though, because statutory parental leave is unpaid.
It is important to remember that you do have the right to refuse a holiday request, whether for paid holiday or unpaid holiday.