The couple, hailing from Villupuram in Tamil Nadu, had been coming to the shrine for the last three years and were making a living by selling beads and chains for devotees, they said.
On noticing their child missing they lodged a complaint with police who scanned CCTV footage near the spot, which showed their child being carried away by a woman at around 3am.
Police also examined footage from surveillance cameras throughout the complex, which captured the woman with and without the child.
Photographs and footage were released to the media and posted on the social media as well, police said, adding that a hunt was on to nab the woman.
In December 29 2018, an 18-month-old boy was stolen from a sleeping Maharashtrian couple on the hills by a man.
He was arrested a few days later from Nanded in Maharashtra and the boy was rescued.
In 2017, two such incidents took place on the holy hills. The abductors were later arrested from Telangana and Tamil Nadu.