Jay, your parents would need to call customer service to troubleshoot the dual tuners. The DirecTV DVR should allow you to record one and watch another, you are correct. But somewhere along the line your receiver isn't getting the second line to allow that.
You can also see about jarring it into working, though, if you are comfortable with electronics and cabling. I can't guarantee it will work, but sometimes it does.
First, make sure that you have to SAT-IN cables plugged it (they're the one that would come from the dish, usually through the wall). Very rarely you would have one, and if there's just one I would leave it to the Tech Support agents to determine if you have a special installation.
If you have both cables, though, try unplugging the receiver from the power outlet, and then unplugging both SAT-IN cables. Leave it that way for about 30 seconds, then plug in JUST THE POWER OUTLET. Wait for it to load (depending on the type of receiver you have, this can take anywhere from 1-5 minutes). It should get to a screen that should have the programming and says Searching For Satellite Signal (771) in the blue box on the bottom. Plug the SAT-IN cables back in and check to see if it will then record two shows at once.
Sometimes this will get the receiver to work correctly. Sometimes it won't though. Other problems could be that one of the cables is damaged, loose, or even too long. Another cause (very rare, but it does happen) could be that the receiver's second tuner isn't working properly.