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).

Mathematician was the top-rated job according to a study discussed in the Wall Street Journal in 2009. Number 2 was Actuary and Number 3 was Statistician. Income for mathematicians was a bit higher than either of those (it surprised me how well-paid "mathematicians" are). Computer-related careers also placed well in this ranking.

Like they say "it's nice work if you can get it"! For some sense of the available jobs, check mathjobs.org. A PhD will likely be required for many but not all.

Regarding costs: You don't need much equipment, so that could help keep costs down. ✎ Scholarships are often available for advanced degrees in this area, so earning a PhD may be more a matter of time and effort than money.

If you're interested in going to grad school for mathematics in the US, have a look at the Mathematics GRE Subject Test, since your results on that test will (likely) be a strong factor in influencing where you are accepted.

The answer very much depends on what kind of mathematics you study. Some purely theoretical topics may lead only to academic positions, whereas other areas could lead to positions in finance, data science, statistics.