No, they can’t use the same cable.
You say your parents have Dish TV already… so what they have is a satellite dish on their roof (or wherever) and the coax cables run from that dish into the house (from what I can tell they usually just drill right through the wall into the room).
Any chance the house was originally wired with coax in the walls? If so, you can use those cables for Comcast as they aren’t used by Dish. If you don’t have any coax in the house at all (aside from what is used by Dish) then your only choice is to have something installed.
I certainly wouldn’t have Comcast do it as I don’t think they do a good job with anything and they’re too expensive, but that’s just my opinion. Perhaps you could convince them to run a cable for you for free as part of the installation? That’s the only way I use Comcast for this. Otherwise I do it myself or find a local handyman to do it for you. You should only need one cable run, as the only thing it’s needed for is to bring the signal to the modem. After that it’s all CAT networking cable or wireless, not coax.