I worked in healthcare settings for around 9 years. Unfortunately, getting exposed to disease is an occupational hazard in that environment.
A lot of this answer depends on your parents' fears and their reaction to them. Either they aren't reasonable WRT their fears, or they are. I'll try to answer both perspectives.
If they aren't reasonable, there's no amount of convincing that you can do. That's part of being unreasonable. In this case, with people, I've just said, "I love this job and I'm helping the sick every day; being exposed is just part of the risk I accept when I come to work." Don't argue, don't try to convince, don't say anything more. Let your passion for your calling in life be the answer to unreasonable fears.
For people that are more reasonable, then you can talk about infection control. There I've said things like "This is just one of many things I risk exposure to every day. The facility does containment of communicable diseases and does it well; the most effective thing to do is to wash my hands and I do that often. Yes, something may happen but do you want to know the statistically riskiest part of my job? It's the commute to work. I stand a lot worse chance of dying in a traffic accident on the way to work than from this particular virus." (Look it up... death in an auto accident, at least in the US, is much more common than other forms of premature death. One source is here: https://en.wikipedia.org/wiki/Motor_vehicle_fatality_rate_in_U.S._by_year)
Either way, those of us who work or have worked in a healthcare setting take on this risk on a regular basis. I know I was sick more often when I worked in the hospital - it can't be helped when you're around sick people. But that was all colds; it wasn't anything serious (TB, flu, SARS, etc.) The key here, especially with the panic going on, is for us as professionals to stay calm. Let others see how calm you are and derive strength from you.