Bakiyev had led protests against the February and March elections of the parliament, triggering the coup. But after his appointment, Bakiyev extended an olive branch, calling for support of the new parliament.
The Supreme Court had reinstated the old parliament on Thursday when allegations of fraudulent balloting led to protests and Akayev's ouster, but the new parliament continued to assert its legitimacy.
"I can be reproached for saying earlier that the (February and March) polls were not legitimate. I said so. But in this parliament we have questions to only 15 to 20 constituencies, no one is saying that all deputies have to go," said Bakiyev.
"In accordance with the constitution, the (previous) two-chamber parliament should finish its work," he said, according to the Associated Press. "The old parliament needs to think about its voters' interests and not about its own."
Bakiyev also urged Akayev, in power since 1990 when the Central Asian country was still a Soviet republic, to resign.
On Monday, Akayev issued a statement to Kyrgyz news agency Kabar from his exile in Russia, accusing the new leaders of disgracing the country and harming the economy, Reuters reported.
He did not indicate whether he would resign or try to return home as he has hinted.
Kyrgyzstan has set June 26 for a new presidential election, although the date has yet to be confirmed.