The east-based House of Representatives said that Fathi Bashagha's government officially started working from the city of Sirte on Tuesday.
Parliament Speaker Aguila Saleh claimed that Bashagha's government can't enter the capital "as Tripoli is under the control of armed groups."
"The government officially decided to assume its duties from Sirte in order to avoid bloodshed," he added in a speech to parliament.
The parliament speaker said the government of Abdul Hamid Dbeibah refuses to hand over government premises in Tripoli to Bashagha's cabinet.
"This is a clear violation of the constitution and law," he alleged.
There was no comment from the Dbeibah government on Saleh's statements.
For more than two months, there have been two governments in Libya: The national unity government led by Dbeibah and the one granted confidence in early March by the Tobruk-based House of Representatives.
Dbeibah has previously said that he would only cede authority to a government that comes through an "elected parliament," raising fears that the oil-rich country could slip back into civil war.
Parliament declared that Dbeibah's term expired when the December election did not take place as planned, and the chamber has instead chosen Bashagha to lead a new transition with elections to follow next year.
Parliament's position is backed by the eastern-based putschist Gen. Khalifa Haftar, who waged a 14-month war on Tripoli from 2019 to 2020. Armed factions in the capital and western regions appear divided over the crisis, with some saying on Tuesday they opposed parliament's move to install a new government.
Bashagha, a former interior minister, said he was committed to holding elections within the time frame next year set out by parliament, adding that he wanted to achieve agreement between rival political institutions on the issue.
Disputes over basic rules for the election led to the collapse of the planned vote in December.
