Dear Physician vs Math-Guru:
We are very impressed indeed that you are in fact considering your choices carefully - unlike many youngsters in India, and elsewhere, who blindly do what they are told or follow the latest fad.
Education is a means to an end, and not an end unto itself. We are confident that you and your parents have the same thing in mind: for you to have a happy and successful career and life. Although you all have the same objectives, the challenge is for you to choose the path that would get you there. No single path is the right path. In fact the entire path does not need to be defined right now, but the selection of the first leg of the path can constrain your future choices.
Life is about a balance - a balance between following your passion and making necessary sacrifices to have a meaningful, rewarding, and comfortable living. The extremes are not likely to produce happy outcomes: following your passion while ignoring financial rewards, or pursuing financial rewards while sacrificing the joys of pursuing your passion. The balance between the two extremes is a personal choice. It is a balance determined by you that would allow you to pursue what you like, while making a reasonable living in the future.
Here is a short story of an individual who found himself in a situation similar to that of yours. Due to parental encouragement and to societal definition of medicine being a superior education, he entered Pre-Med in India after completing high school. Within weeks, he found that he disliked medicine and dropped out to enter engineering. He ended up in Canada and went on to do his PhD in electrical engineering. He became a university professor in the US, only to realize later that he wanted to be in a business, and not in an academic, environment. He left to get an MBA and became a management consultant. And, then he was doing what he really enjoyed - advising senior executives on how to improve their business. Now, he is a founder of iMahal. According to him, his life is as rich and enjoyable as that of any other professional, even that of a physician.
Pursuing medicine at this stage has a built-in assumption of future professional success, granted that this may not be your definition of a successful life. Pursuing mathematics from a prestigious institution like Indian Institute of Technology (IIT) isn't a pushover either. However, what career you would pursue subsequent to your degree isn't as clear cut as it is for studying medicine. Your parents' concern is well-placed, which is: you might study something in the short term that you enjoy, but you may end up in jobs for the rest of your life that are neither rewarding nor enriching.
If you pursue mathematics, you should honesty consider what it is you would do with your hard-won knowledge. After your first degree in mathematics, you can seek employment or pursue further education. Depending on your specialization in higher education, you can gain employment in academia and industry. Should you choose to explore other areas of education after your first degree, you could qualify for a Master's in computer science or an MBA.
It is your choice to go for a "sure thing" with professional success in medicine, or pursue education in mathematics with a yet-to-be-defined and somewhat ambiguous career path. It is your personal choice as to which one of the two is better. Being excellent at something and also enjoying it is half the battle in being successful. Carefully assessing your options at your young age proves that you are well on your way, no matter what you choose.
One final piece of advice: talk to some physicians and some mathematicians, ideally those who are as old as your parents and who do not know your parents. Ask them how they feel about how they make their living. This takes some fortitude, but you will be proud of having faced your future with your head held high.