Do I need to plaster plasterboard on bathroom ceiling?

Hi. While not essential that it was plastered, it is preferable. What you now have is a very porous surface (especially the dry-lining filler used in the joints) i never advocate the use of pva on any surface that is to be painted and consider advice to do so bad advice. You can purchase and use a dry-wall sealer, or paint with paint for new plaster, or an ordinary emulsion thinned with water 50/50 first coat and diluted less on subsequent coats. These methods allow priming of the plasterboard and reduce its porosity. As its a bathroom ceiling you need to carefully consider the finishing paint. Bathroom paint as you say or what i would use if the job were for a customer is zinnsser permawhite, 2 coats of this paint to finish will give the ceiling the best protection from the enviroment of a bathroom.

Hopefully they used green board. Yes, you can sealed and paint. Screws & joints will need to be taped and jointed. Me personally I would plaster the ceiling.