Is a divorced woman living in her parents' house obligated to seek her father's permission to go out, or is it sufficient to inform her mother?

There is no harm in a girl going out to fulfill her needs, as long as she is covered and modest. She is not required to seek her father's permission for every outing unless there is a risk involved. If the father does not prevent her from going out for dawa (preaching), visiting relatives, or visiting the sick, then she may do so with reservation, chastity, covering, and hijab, even if she does not seek her father's permission, unless there is a risk in her going out, in which case her father's permission is necessary.