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.
Al-Masirah, the TV station of the armed Houthi movement, said on its Twitter account that 39 people had been killed and 51 wounded.
The Yemeni Health Ministry said at least 50 civilians, mostly children, lost their lives and around 77 others were wounded in the attack, which Qassemi described as a "war crime".
The Save the Children charity, quoting its staff, said that at the time of the attack the children were on a bus heading back to school "from a picnic when the driver stopped to get a drink".
On Wednesday night, Col. Al-Maliki stated that at (20:34) the Coalition Air Defense detected a ballistic missile launch from Northern Amran in Yemen, aimed at Saudi Arabia.
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Earlier, Maliki accused the Houthis of "recruiting child soldiers, throwing them in battlefields and using them as tools".
It was not immediately clear whether the bus was the target of the air strike, but coalition spokesman Col Turki al-Malki said the attack was "a legitimate military action, conducted in conformity with global humanitarian law".
Abu Muzahem, a resident of Dahyan, was working building a house when the air strike hit the market on Thursday morning.
On Wednesday, the official Saudi Press Agency carried a statement by the Saudi-led coalition saying fragments of a missile fired by the Houthis into the kingdom's south killed one civilian and wounded 11.
The missile was sacked from the rebel-held Yemeni province of Amran towards the Saudi city of Jizan, the coalition said.
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The Saudi government issued a statement on August 9 and described the strike as "a legitimate military action" and was "conducted in conformity with the global humanitarian law".
Col. Al-Maliki stressed: "The Joint Forces Command of the Coalition will take all deterrent measures against such barbaric, frivolous launches in conformity with the worldwide humanitarian law, and those who support these terrorist crimes will be held accountable for their actions".
The Houthis have in recent months ramped up missile attacks against Saudi Arabia, which Riyadh usually says it intercepts.
The United States called on the Saudi-led coalition to launch an investigation into the events that unfolded in Yemen on Thursday.
Yemen's war has left almost 10,000 people dead since 2015 and unleashed what the United Nations describes as the world's worst humanitarian crisis. The coalition denied that it had carried out air strikes in the area, and blamed the deaths on rebel mortar fire.
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"It's hard to believe we live in a world where children should live in fear of such attacks, yet here we are", she added. "Parties to the conflict and those who have influence over them, including Security Council members, can and should choose to end this catastrophe for the sake of Yemen's children".