"Can a Minister remain in his role while not a member of parliament? Section 64 of the constitution seems to imply they can for three months."

You are right John. Section 64 of the Australian Constitution gives ministers three months to gain a seat in either the Senate or the House of Representatives before they have to give up their job.

When a seat in the House of Representatives is made vacant, it is filled by a special election for only that seat called a by-election. Section 64 allows a minister to continue with their job while they are standing in a by-election.

The situation is different for senators. Recently a senator who was also a minister was found by the High Court of Australia to have been not properly (invalidly) elected. After she was removed from Parliament, the minister resigned her ministerial commission. Because there are no by-elections for the Senate, if the former minister wished to return to the senate, she would have to stand at the next senate election. 

diagram to help explain Can a Minister remain in his role while not a member of parliament? Section 64 of the constitution seems to imply they can for three months.