I'm thinking of not moving out of public housing after getting a new job
Will you be forced out if you refuse to move yourself?
No but my parents want me to.
I understand, it's probably easing the issue of living paycheck to paycheck. Tell your parents that your situation isn't quite relieved and you need more time to settle in.
I don't want to be scolded.
You need to be able to afford a place to live and if you are able to stay then you should. You don't need to be broke and struggling if you just got a new job.