Johannes Bruwer, head of delegation for the ICRC in Yemen, said: "Scores killed, even more injured, most under the age of ten".
The bus hit by the missile was ferrying local civilians, a majority of them children, according to Yemeni tribal leaders.
The move comes as part of a larger sweep to disarm the east of Yemen and clear the area of Al Qaeda and ISIS.
Fragments from the Houthi missile launched at Jizan Industrial City had killed one Yemeni civilian and wounded 11, Saudi state media said earlier on Thursday.
Saudi Arabia and Sunni Muslim allies have been fighting in Yemen for more than three years against the Houthis, who control much of north Yemen including the capital Sanaa and drove the government into exile in 2014.
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Last week, the Saudi-led coalition airstrikes struck the gate of al-Thawra hospital and adjacent fish market in Yemen's Red Sea port city of Hodeidah, killing 52 civilians and wounding 102 others.
Turki al-Maliki said the kingdom's air defense had intercepted a ballistic missile fired by the Houthi rebels at a densely populated civilian area in Jizan, according to the Saudi Press Agency.
Al Masirah TV gave a different casualty toll, saying 39 people were killed and 51 wounded, mainly children.
"Under the global humanitarian law, civilians must be protected during conflict", it added.
Later on Thursday, airstrikes hit the Yemeni capital, Sanaa, and sounds of the blasts reverberated across the city's southern and western neighborhoods. That assistance has come under sharp criticism from Congress and the worldwide community as civilian deaths have continued to multiply, even as the coalition promises not to target civilians.
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It has launched thousands of air strikes in a campaign to restore the internationally recognised government of President Abdrabbuh Mansour Hadi.
Sen. Chris Murphy (D-Conn.), an outspoken critic of the United States' support for the Saudi coalition, expressed fury over the attack and demanded once again that lawmakers end their complicity in the war. The simultaneous power vacuum and crippling of Yemen's already-weak health and security infrastructures have enabled al Qaeda and ISIS affiliates to operate, albeit under threat of USA airstrike, and unleashed a tsunami of starvation and disease. But the coalition denied carrying out any attacks in the city, saying it follows a "strict and transparent approach based on the rules global law".
The attack on Jazan was the second ballistic missile fired at Saudi Arabia by the Iran-backed rebels this week.
The Yemen war has killed more than 10,000 people, displaced more than 2 million and driven the country to the verge of starvation, according to the United Nations. They say their missile attacks on the kingdom are in retaliation for air raids on Yemen by the Western-backed coalition. He recently announced plans to invite Yemen's warring parties to Geneva on September 6 to hold the first round of consultations.
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