The coalition denied responsibility for those attacks.
Houthi-run Al-Masirah TV reported that 47 people were killed and 77 wounded, and broadcast graphic pictures showing the bodies of several young children, some of them wearing school uniform.
A Reuters photographer saw bloodied and bandaged children being treated by doctors.
The children were on their way to summer camps, Yahya Sha'em, head of the Houthi-held health office in Saada told CNN. Most of the missiles have been intercepted by the Saudi military.More news: Female heart attack patients fare better if treated by a woman
In 2014, the Houthis overran the Yemeni capital and seized control of much of northern Yemen as well as a string of ports on the Red Sea.
The head of the humanitarian organisation's delegation in the country, Johannes Bruwer, later tweeted that "scores" had been killed and "even more injured, most under the age of 10".
"Shrapnel from the intercepted missile scattered over residential areas, leaving a Yemeni resident killed and 11 other civilians wounded", the coalition said.
The ICRC stressed that "under global humanitarian law, civilians must be protected during conflict", while the secretary-general of the Norwegian Refugee Council Jan Egeland called it a "grotesque, shameful" attack that showed "blatant disregard for rules of war".More news: Iconic 'Brady Bunch' house purchased by HGTV
The military alliance says it does not target civilians and has set up an investigation committee into alleged mass casualty air strikes which have mostly cleared the coalition of any blame.
"Following an attack this morning on a bus driving children in Dahyan Market, northern Saada, (an ICRC-supported) hospital has received dozens of dead and wounded", the organization said on Twitter without giving more details.
The Houthis have launched a series of missile strikes on the kingdom, including Riyadh, over the past year.
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.More news: Bashar al-Assad's wife being treated for breast cancer
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