I've mailed cash before. Some few times that I have for most people it DID get there just fine. For another person who sent me some plants (aquatic) it didn't get there, twice. Now, I have no reason to believe that this person would try to scam an extra $5 out of me, so I have to believe it was someone handling the mail on her end (on mine, I've never had a problem). Lesson learned, though, cash through the mail is a VERY bad idea.
Ditto on the identity theft. It's FAR more likely that some unscrupulous employee at location X is randomly saving and using or selling the information gained. It's also usually not your id that's stolen, but just your CC (or debit card) number, expiration, name, and CVC. This recently happened to the owner of the company I work for. He was on vacation with his family. As it happens, when they're out, they usually use HIS debit card. Except one morning he had to be out early so his wife took the kids to the same restaurant that they'd been to a day or so before. She used her card for the first time in months, and didn't use it again after that. He happens to check and noticed the few dollar and dime charges before anything bad happened. The thing is, they were doubled, one for each debit card, for services (usually porn site access or something like that), not goods to be shipped. This is how it usually happens, first a charge of a few cents, then a few dollars, then maybe ten or twenty dollars, then the sky's the limit. Another reason why monitoring your accounts PERSONALLY is so important. I've had it happen twice already. The second time, I knew exactly which location was responsible (long story) and went back and complained. Of course, they insisted none of their employees would ever do that. but, as it happened, a few other people in my area also had the problem at around the same time and ALL had been to this particular place within days of it happening. Anyway, long diatribe short, your parents are in more danger of CC theft using their cards face to face than they are, really, online, but nothing you say can, nor should it, change their minds. It's their money and their prerogative to do what they will. So, I'd look at alternatives, like gift cards, before I'd try to talk them into letting you use their cards online.