Whether or not you pay tax on your part-time job depends on how much you earn, not on the number of hours you work. Everyone receives a certain amount of income in each tax year on which no tax has to be paid. This is called the Personal Allowance (£8,105 in 2012-2013). If your earnings from your part-time job are below this, then you do not have to pay tax on them. If your earnings are more than this, you will pay tax on the difference.
One advantage to having multiple part time jobs is National Insurance, each job has its own allowance which means you end up not paying any NI (and so could your employer). However, there is a risk of aggregation (treated as from one source) if the jobs are related.
Students often work during the holidays and then return to full time education and use form P50 to reclaim their tax withour waiting till the end of the tax year, here is a link to P50
You don’t have to be a student to use P50 it can be used by anyone stopping work or retiring.