I thought they were found in Peru, in the Ica province. Hence their name. The driest desert in the world (i.e., lowest rain levels) is the Atacama Desert in Chile. Or has climate changed revised the world's record for deserts? (I hadn't thought about that before but it sounds possible.)
Rain is not the only source for erosion. Various natural processes cause erosion, such as wind, sunlight exposure, chemical reactions (such as routine oxidation over time), and even animal movements. The term erosion covers far greater ground than you are presuming.
Anyway, why would there be large dinosaurs living in a desert? Or are you saying that that desert was tropical and only recently became a desert? (Considering the climate patterns which produced that desert--a result of the obstructive mountains shielding that area from rain clouds---that would be a very bold claim.) Or are you assuming that the culture which produced the Ica stones lived in the desert even though the dinosaurs lived elsewhere?
I'm still unclear how you know that the "dinosaur" drawings couldn't be based on mythical stories about monsters? (However, one of the stones I recall seeing looks more like a kid's drawing of Barney the Purple Dinosaur.) Of course, for the moment I'm simply temporarily assuming that the stones are indeed from an ancient culture, even though the weight of the evidence defies that idea.
P.S. I keep recalling a fun cartoon from long ago---which only has humorous reference to our present topic--- where it shows scientists examining petroglyphs in some cave. They posit very sophisticated theories about the meanings of the drawings and the profound worldviews they represented. And they try to decipher the meaning of the strange symbols next to the drawings. Then the final frame is introduced with the caption, "Meanwhile, a flashback to the same cave over 50,000 years ago...." One little caveman kid says to the other while all of the adults are sleeping around the campfire, "I can draw a monster far scarier than your monster! Watch me. And these little marks are the sticks the monsters eat for food!" (As I said, it has nothing to do with Ica stones but the cartoon keeps coming to mind.)
One thing is for sure: This thread has been a very active one. Already nearly 150 comments posted! Dinosaurs and mysteries tend to get attention from us humans.