my friend frequently brings her dog to stay on my cats' catio (without the cats in it). if her dog were to have worms or parasites, could her dog transfer those somehow to my cats via having been in the catio?
Yes the cats can get worms or parasites (like fleas or ticks) from a dog that was in the same area and is infected. The fleas or ticks for example could just simply fall or jump off the dog. Worms can be transferred by poop (an infected animal's poop contain eggs), so if the dog poops in the area or has poop on it's feet, and it's not completely cleaned up. Not just picked up but the spot cleaned with cleaner to kill any worm eggs. The cat could sit or step in the area and pick worm eggs up and swallow them while grooming.
my catio is a screened in, tiled area that i can close the door while the dog is there. my fear is that the dog lives in the country where there are barn cats that poop "wherever" and the dog often eats the cat poop (yuk) and i'm sure she steps in it as well. she gets her monthly flea/tick med but nothing for (potential) worms. i don't want to hurt my friend's feelings by telling her i can't do "doggy day care" any more (she comes 2 or 3 times a week while my friend runs errands because she doesn't want to leave the dog at home, but i don't want my cats picking up anything either.
Just tell her that you don't want to risk your cats getting sick, so if she wants to continue to be able to bring her dog over it needs to be regularly dewormed. The other option is to just to a through cleaning of the catio after the dog leaves every time before allowing the cats back in there.
I just noticed you are in Texas? I don't really know about the mosquito situation there, so I don't know if heartworm is an issue, but if it is, does the dog get monthly heartworm meds? If so it's likely already getting dewormer as most of the heartworm meds work for other common worms too.