Usually, when you arrive in Korea as an international student, you get either a D-2 or D-4 visa to study. D-2 and D-4 visa holder students can work in some specific part-time jobs as long as they are allowed by the University, at which they are enrolled.
All foreign students holding D-2 visas can only work part-time jobs after applying for an S-3 (part-time employment) visa. With an S-3 visa, undergraduate students can work 25 hours per week while master or doctoral degree and research students can work 30 hours per week. Students must work in permissible fields such as restaurants, shops or academies. Moreover, some schools can offer job positions in their institution, like language exchange programs where students are paid.
To get an S-3 visa you have to submit the following documents at the immigration office:
Alien Registration Card
Part-time Employment Confirmation Form
A letter of recommendation for part-time work
Business Registration Document
For students holding a D-4 visa, if they want to work, they must study in a Korean university for at least 6 months which means to start work at a part-time job you need to be in Korea for more than 6 months. After that, they can work 20 hours a month during the semester and during vacation, there is no particular limit.
To get more info and reserve an appointment at the immigration office please refer to the following link: https://www.gov.kr/portal/foreigner/en/m020103
KGSP Students Case:
Scholars who received any warning and scholars enrolled in the Korean language program are strictly not permitted to hold part-time jobs.
Also, scholars are not allowed to take a part-time job during the academic semester.