If you're a domestic student you can study for a master's or PhD part-time. You'll need to provide the reasons why you need to study part-time when applying for admission.
If you're an international student, check with the Student Visa Services team at the International office (email student.visa@otago.ac.nz) to determine if you're able to study part-time. Typically, a student visa does not permit part-time study.