You are only allowed to work part-time not within study hours. The number of your working hours cannot be less than fifteen (15) hours per week or their equivalent in days or months per year.
In this case, you will fall under the provisions of labor law (ie, you must pay social security contributions, submit a tax return, etc.).
Two types of residence permits cannot be granted simultaneously to a citizen of a third country.
Nevertheless, you may exceptionally be granted a student residence permit if you reside in Greece as an employee or as a self-employed person as long as you were previously accepted for reasons of family reunification and held a relevant residence permit. In this case you will continue to stay in Greece as a student only with a student residence permit.