While one of them has been endorsed by the Supreme Court, the other has the Election Commission's blessings.
This deadlock comes after massive street protests over the first round of February 27 parliamentary elections, preliminary results for which gave President Askar Akayev's party a huge lead and his son Aidar a seat in parliament.
International observers said the vote fell short of international standards, and was conducted in a tense atmosphere and that the independent media was muzzled.
The rumblings of dissent spilt on to the streets after a March 13
On Thursday, they stormed the presidential headquarters, forcing Akayev to flee to Moscow.
A day later, the Supreme Court annulled the elections and restored the earlier parliament as an interim government tasked with fresh elections.
But the lawmakers elected in the last elections insist that they have the mandate to rule, and the two sides met at separate chambers in the government buildings to chalk out future strategy over the weekend.
A delegation from the Organization for Security and Cooperation in Europe has landed in Bishkek to help resolve the deadlock.