Sunni-ruled Saudi Arabia and its allies, including the U.S., see the Shiite Houthis as being financially and militarily supported by Shiite-ruled Iran.
Iran denies this, although the U.S. Navy says it has intercepted several shipping boats since the war began carrying Iranian weaponry suspected to be on the way to Yemen.
Iran is a regional foe of Saudi Arabia and continues to be a critic of the government in Riyadh and its bombing campaign in Yemen.
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However, the Washington Institute for Near East Policy said there were limits to Iran's ties with the Houthis.
"Houthi relations with the Islamic republic resemble the Iran-Hamas relationship more than the Iran-Hezbollah relationship — that is, the Houthis are autonomous partners who usually act in accordance with their own interests, though often with smuggled Iranian arms and other indirect help," analysts wrote in a report early Thursday.
Saudi Arabia and Egypt fear the civil war threatens the globally-important Gulf of Aden and the Suez Canal because Yemen sits on the narrow Bab el-Mandeb strait through which much of the world's oil shipments pass.
Recent missile attacks have underscored those fears. An Emirati-leased Swift boat came under rocket fire from Houthis in the area last week, suffering serious damage. The United Arab Emirates described the vessel as carrying humanitarian aid and having a crew of civilians, but Houthis said it was being used as a warship.
The Houthis on Thursday reiterated a denial that they carried out the failed missile attacks on the U.S. navy destroyer, a news agency controlled by the group reported, according to Reuters.
They said the missiles did not come from areas under its control, said the Saba news agency, citing what it called a military source.
"These allegations are unfounded and [Houthis] have nothing to do with this action," the agency reported the source as saying. “Such claims are part of the general context of creating false justifications to escalate assaults and cover up the continuous crimes committed by the aggression against the Yemeni people, along with the blockade imposed on it, and after the increasing condemnations to such barbaric and hideous crimes against Yemenis."
Unemployment, high food prices and limited social services mean more than 10 million Yemenis are believed to be food insecure.
With food ships finding it hard to get into Yemen's ports due to a virtual blockade by Saudi Arabia, over half the country's 28 million people already do not have enough to eat, according to the United Nations.
The World Food Programme says half Yemen's children under five are stunted, meaning they are too short for their age because of chronic malnutrition.
Alastair Jamieson is a London-based reporter, editor and homepage producer for NBC News.