How should I convince my parents about a career in mathematics?
Your parents are thoroughly incorrect when they say that there is no career opportunities in mathematics. 

Further to Dave Clarke's answer, many (applied) mathematicians find work in the City (as in the financial services sector of London) running risk analyses, for instance. 

If however, you are looking at finding work as a research academic in mathematics, then jobs are there, but sought after. Teaching mathematics at university level provides wider prospects, however. 

To answer your question about money, the amount you have to pay largely rests on the tuition fees that you can afford. This will depend on which university you choose to attend.

There is also the middle ground of studying mathematics with a view to putting that study to work in medicine. Consider that statistics (yes, probably not the subject you are interested in), was essentially developed as a response to a medical situation (plague in London). There is also a vast amount of work requiring pure and applied mathematics in medicine (nuclear medicine, MRI, NMR, for example).