做銀行工作可以另外做part-time嗎?

Yes and no. It is all depended on the employment contract between you and your employer. Most likely, contract has a clause stating that you have to report for any other employment you are involving. If that clause is on your employee contract, then you are surely breaching contract.

2008-04-16 23:28:21 補充：
Basically, I has been working in HSBC for a quite a while, but I am still keeping my part-time job as private tutor.
I believe the reason why you heard about the part-time thing is because information you involve in working in bank is very sensitive.