My parents work in a textile factory built by themselves at home, so are they self-employed? What should I fill in the work unit and position?

If you build a factory at home, and handle the industrial and commercial registration of individual industrial and commercial households, and get an individual business license, it is an individual industrial and commercial household. There is no problem in filling in individual industrial and commercial households in work units and posts. If you don't get an individual business license, you can't be a self-employed person, but you can write a self-employed person.