Please don't underestimate how much time to take the children out of school,
I suspect that you will find that you will not want to leave your dad, I had to leave quite quickly (couple of days after mum died) , I really regret it, because Dad really need me there,
In the run up to the funeral, time is structured, and there are tasks to attend to, it hit Dad hard afterwards, as well as myself and siblings,
I am sorry that this head teacher, has such limited life experience, that she has added to your stress,
Do not allow anyone to pressurise you into any time limits, it is really important that you give yourself what ever time you need in which ever location is appropriate.
Make sure that all the staff at the school get appropriate training, in order to prevent any future inappropriate actions, contact the local Cruse Bereavement Care they will come in and do a training day for the staff, as they should not be this unaware.