One grenade was believed to have been fired from a flyover at a medical tent and another one at the women's toilets nearby, reports said.
The incidents took place at 8.26pm.
The site of the People's Democratic Reform Committee on the road is under leadership of Luang Pu Buddha Issara. It is the only stage outside Bangkok.
It was the first time that the Chaeng Wattana stage had been attacked with M79 grenades. The explosive devices were fired at a stage in Lat Phrao last week, prompting anti-government leaders to close down the stage and another one at the Victory Monument.