The two groups are fighting a Saudi-led coalition that has intervened in a 2-1/2 year Yemeni civil war with a view to restoring the internationally recognized government of President Abd-Rabbu Mansour Hadi.
"We vow to our brothers and neighbours that, after a ceasefire is in place and the blockade is lifted. we will hold dialogue directly through the legitimate authority represented by our parliament", Saleh added.
Former Yemeni President Ali Abdullah Saleh - whose supporters are now in a shaky alliance with the Houthi rebels in the fight against the Saudi-led coalition - tossed an olive branch on Saturday to Riyadh, saying he is open to talks with the coalition and ready to "turn the page".
The United Nations has warned that the country faces mass starvation unless the Saudi-led coalition allows more food aid to enter the impoverished country.
A Saudi-led coalition against the rebels imposed the blockade, which the United Nations warns is deepening Yemen's humanitarian crisis, in response to a Houthi missile that was intercepted near Riyadh airport on November 4.
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Saleh's supporters managed on Saturday to seize control of several embassies and government buildings from the Houthis.
"I call upon the brothers in neighbouring states and the alliance to stop their aggression, lift the siege, open the airports and allow food aid and the saving of the wounded and we will turn a new page by virtue of our neighbourliness", Mr Saleh announced in a televised interview with the TV station Yemen al-Youm.
Pointing to global law, Lowcock said, "Wars have rules, and they need to be complied with".
"The decision by (Saleh's) General People's Congress to take the lead and their choice to side with their people will free Yemen of. militias loyal to Iran", the statement carried by the official Saudi Press Agency said.
"It is not unusual or surprising that Saleh turns back on a partnership he never believed in", the group's political bureau said in a statement.
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The Houthis' Ansarullah group, in a statement issued on its Twitter account, described the clashes as "regrettable".
For his part, Houthi leader Abdel Malek al-Houthi called on Saleh to end what he called "the sectarian strife" that he has instigated, and threatened to take action against his supporters militarily if he does not.
Saleh's GPC party accused the Houthis of failing to honor the truce and claimed in a statement on its website that the Houthis bear responsibility for dragging the country into a civil war.
In March 2015, warplanes from Saudi Arabia, the UAE, Kuwait, Bahrain, Qatar and other allies began airstrikes on rebel positions in Yemen.
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