The monks plan to spend the night outside the building and to petition the drafters on Thursday.
About 2,500 monks and their supporters had gathered outside the building by early afternoon, awaiting the arrival of the marching elephants.
Bangkok's police had urged the monks not to bring the elephants into the city, for fear of causing traffic accidents or that the heat would irritate the animals and spark a rampage.
After the march began, police relented and allowed the monks to lead the elephants through the streets.
The military, which seized power in a coup in September, has appointed a panel to write a new charter for Thailand.
The first draft was released last week for a period of consultation, before going to the public in a referendum in September.
The publication made no mention of a national religion, but Surayud Chalanont, the army-installed prime minister, said on Wednesday that his government would consider the monks' demands.
Surayud said: "There is room for discussion. The government will listen to people's opinions, and pass them to the Constitutional Drafting Committee."
About 95 per cent of Thais are Buddhist and the military government has also said it would not object to naming Buddhism as the state religion.
Critics of the proposal warn that naming a national religion could inflame tensions in the Muslim-majority south of Thailand, where a separatist battle has raged for three years.