Q: I am hoping you may be able to clarify for me the following:
All employees (including part-time and casual employees) are entitled to 10 days’ sick leave per year if:
they have six months’ current continuous employment with the same employer
What is classified as continuous employment?

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A: There are two criteria to get sick leave (also applies to Bereavement and Family Violence), not just the 6 months:

Section 63 Entitlement to sick leave and bereavement leave

Here is the reference from IRD (Page 43):

Section 63 Entitlement to sick leave and bereavement leave
(1)
An employee is entitled to sick leave and bereavement leave in accordance with this subpart—
(a)
after the employee has completed 6 months’ current continuous employment with the employer; or
(b)
if, in the case of an employee to whom subsection (1)(a) does not apply, the employee has, over a period of 6 months, worked for the employer for—
(i)
at least an average of 10 hours a week during that period; and
(ii)
no less than 1 hour in every week during that period or no less than 40 hours in every month during that period.
So, you must apply both to see if either fit the work pattern of the casual employee.