Houthi-controlled television reported new Saudi air strikes on Thursday in Hajjah province close to the Saudi border near the Red Sea coast, and in the vicinity of Taiz further inland to the south.
Saudi Arabia's civil defence said a Yemeni living in Saudi Arabia had been killed by debris from a drone intercepted over the western city of Taif, the first death Saudi officials have announced in repeated Houthi attacks since fighting escalated last week.
It has previously said more than 80 people were wounded.
In Djibouti on the African shore of the Red Sea, Wahida Omer, 50, reached land after fleeing on a small boat across the Bab el-Mandeb Strait from Yemen when her home village of Dhubab came under attack as the Houthis swept into the area.
Her husband had been left behind and she had no idea if he was alive, dead or captured by the Houthis.
"The sea was frightening, difficult," she said in an interview filmed by the International Organisation for Migration.
Those on the boat were "crying the whole way, and we arrived still crying" with nothing to eat.
The IOM says more than 100,000 Yemenis have been displaced by the latest fighting, mostly within Yemen.
The Iran-aligned Houthi rebels, who have seized Yemen's entire Red Sea coast since last week in a lightning advance, say the Saudi Air Force has conducted hundreds of air strikes in recent days.
They in turn have fired missiles and drones at targets across the border.
On Wednesday, the Houthis said they had shot down a Saudi F-15 fighter jet.
Video they released overnight showed fighters shouting "Allahu Akbar" in the charred wreckage of a crash site and holding up a piece of a military aircraft wing bearing Saudi insignia.
"We want the Saudis to cease their blockade and occupation of our country, and to stop their aggression against us. This is our demand!" Houthi leader Abdul Malik al-Houthi said in a televised speech, seated with a traditional dagger in his belt.
Italy's government said on Thursday it will deploy warships to ensure the safe passage of its commercial ships through the Bab el-Mandeb Strait off the coast of Yemen.
The country does not intend to wait for a joint decision by the European Union, Italian Defence Minister Guido Crosetto said.
"We have the capabilities to protect the passage."
Crosetto warned that if the waterway were to become impassable, this would have significant economic consequences.
"We must not allow bureaucratic delays in decision-making to exacerbate an already complex situation."
The Bab el-Mandeb Strait connects the Red Sea with the Gulf of Aden, one of the most important shipping route between Europe and Asia.
The Houthi advance in the region has given them virtually unrestricted access to the strait.
The rebels have said they are targeting only Saudi-linked vessels, and container ships and oil tankers continued to transit the Bab el-Mandeb Strait, according to shipping analytics firm Windward.
with DPA and AP
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