Most part-time jobs are advertised in the same way as full-time jobs and so it is worth checking job advertisement sites such as indeed.co.uk and lovelocaljobs.com.

If you are targeting specific employers, then check out their website page as they may have a careers/jobs page which advertises their latest vacancies.

You can also use a speculative approach and hand in a copy of your cv to shops/business in your area. See section on 'Preparing for Employment'  https://ccgonline.chichester.ac.uk/pluginfile.php/182273/mod_resource/content/0/Preparing%20for%20Work.pdf for further information.