At least ten prehistoric raptor tracks measuring up to 17 inches long have disappeared from Red Fleet State Park in the last six months.
"It's become quite a big problem," said park manager Josh Hansen. "They're just looking to throw rocks off the side. What they don't realise is these rocks they're picking up, they're covered in dinosaur tracks."
One teenager was recently caught hurling slabs of sandstone into the 750-acre reservoir at the site, said Mr Hansen.
When challenged he was holding another slab bearing two toe imprints from a dilophosaurus track.
“Some of the tracks are very distinct to the layperson but just as many are not," Mr Hansen told the Associated Press.
"That is why it is important to not disturb any rocks at the dinosaur trackway.”
Staff have started adding extra signs to try and prevent further vandalism.
The park, which attracts more than 30,000 visitors a year, is also considering sending a diving team into the reservoir to try and recover the footprints.
However it is feared that many have been lost forever, either because they shattered on impact or dissolved in the water.
"Education can play a big part in stopping this kind of behaviour," said Utah State Parks on its blog about the problem.
"To help combat it, we are asking everyone to spread the word. Please, do not throw any rocks in the dinosaur track area at Red Fleet State Park.
"Help us keep the area preserved and beautiful for visitors both tomorrow and for generations to come."
The dilophosaurus, which weighed up to 400kg and is thought to have hunted in packs, is probably best known from the Jurassic Park movie.
In the film it was given an expanding frill to warn predators and the ability to spit venom to a distance of six metres.
More about: Dinosaur