At first I wasn't alarmed: they seem to be lots of brand new customers on Airbnb and a lot of them don't bother to fill out anything in their profile. I just hosted a young woman with her friend, same scenario, practically no info on her, but it worked out great, at least I could read where she had gone to school and she had two pictures of herself up.
But learning of the situation of supposedly booking for parents only when you inquired about the situation and asked for more details, and then all those vague scenarios, possibly staying longer, medical treatment ?, no picture , it all makes me question who they are and what is going on. Personally, I would just decline them - asap!
Declining is not a cancellation, so no harm is done to your account : Peace of mind is worth something! I see no good in engaging into more and more possibly iffy scenarios with them and possibly getting sucked into it if they elaborate on the medical stuff. ( I doubt that is true..)
Not to worry, you will get plenty of guests in the future! You could always tell this person nicely that you declined but that you'd be happy to reconsider if they have filled out their profile with more info, and then reapplied. Chances are you will never hear from them again - and you saved yourself time and aggravation!
I would also consider flagging their communication with you and submitting it to Airbnb.
Good luck, you are doing all the right things! May you get great renters for your next guests 🙂