"The General People's Congress and its allies hold Ansar Allah fully responsible for every drop of blood shed among the Yemenis. and warn against all acts that, rather than serve national unity, threaten our internal unity and cohesion", the party said in a statement.
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. The civilian suffering was exacerbated by the Saudi naval and air blockade of Yemen, human rights groups say.
Saudi Arabia has led coalition of Sunni Arab nations that have been fighting Saleh and the Iranian-backed Houthi rebels since 2015. There is a patchwork of forces, including Al-Qaeda in the Arab Peninsula, Hezbollah, whatever ragtag troops Saudi Arabia brought in like Senegal units. That now has been partially lifted - though not enough to stave off a humanitarian catastrophe, according to the United Nations.
Former Yemeni President Ali Abdullah Saleh said on Saturday he was ready for a "new page" in relations with the Saudi-led coalition fighting in Yemen if it stopped attacks on his country.
"I call upon the brothers in neighboring 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 neighborliness", Saleh said in a televised speech.
Within hours, Houthi spokesman Mohammed Abdulsalam pushed back.
The war in Yemen is one of the worst humanitarian crises in the world, killing at least 10,000 people and leading to widespread hunger and disease.
Three-fourths of Yemenis need some kind of humanitarian assistance to meet basic needs, global observers have said, with more than 17 million people facing food insecurity, including 8.4 million at risk of starvation.
Together they have fought the Saudi-led coalition which intervened in Yemen in 2015 aiming to restore the internationally recognized government of President Abd-Rabbu Mansour Hadi after the Houthis forced him into exile.
The Huthis have also accused the former president of accepting funds from the Saudi-backed Hadi government.
The Houthis, a Shiite tribal militia from northwest Yemen, have been at war with the central government for the better part of a decade.
He added that some speculators claim that Saleh's statements are actually a proposition by Saudi Arabia to roll down the hostilities in a manner which would save face.