You would probably have a work permit for the full-time job at the school, but the part-time work at the language institute would just be for a few hours in the evening.
One of Bangkok's biggest private language schools only employs part-time teachers who already have a work permit for their full-time job. This means that at the very least, the teacher is legally entitled to work in Thailand, albeit not for the part-time job.
However, an ajarn reader got in touch to say that at the back of the work permit book, there is space to add a 'second employer'. So it seems you can possibly cover two jobs at two different locations with just one work permit book.