In the first episode of the anime, humans and birds are shown to have turned into stone. But dogs, fawns, etc. didn't.
Why did humans and birds turn into stone but other animals didn't?
It seems whatever happened was aimed at humans, just like how they show that once it was aimed at only birds, and not humans.
It is probably going to be revealed.
Spoilers for the last few chapters of Dr. STONE:
The source of the initial petrification at the start of the series was the artificially sentient Why-man/Why-men (they acted as a collective so both could fit), who could choose which species they wanted to petrify. First, they tested this on swallows, then later on humans. They did this in order to fulfil their goal of protecting humanity, as they believed that the human race would benefit from becoming stone. They didn't select other species since they believed that humans were the most advanced and intelligent on the planet.