There are plenty of part-time jobs available in Durham and you'll have access to a range of support from the careers team in this process. For starters, they have an online portal where part-time jobs are advertised. Many of these are local ones, but there are also jobs within the uni- eg, within college bars or within the professional service departments like marketing, careers, academic departments, etc. These are relatively easier to secure since they're reserved only for Durham students, so the competition pool is smaller. The careers service also runs a 'part-time jobs and volunteering fair' in Michaelmas term where local employers come on campus and showcase the opportunities they have for students, so I'd recommend attending that.
As for the uni jobs I mentioned above, I have two- one within marketing where I engage with people here on TSR and the other within the Careers department. They've both provided me with amazing work experience alongside being very flexible and paying higher than minimum wage. There's also an opportunity to become a Student Ambassador for Durham- these roles open up at the beginning of the academic year so you can keep an eye out for these on the portal as well.