Stoned Birds are a result of a select number of birds that were affected by the Petrification alongside Humanity.
Certain birds were trapped in stone along with the entire human population of the earth and it is unknown why.  After awakening, Senku used a number of stoned birds for his experiments. He tested a number of cures until, after a year, he set a bird free from petrification with the stone formula. 
All swallows were turned to stone.