Medical sources quoted by the organisation said the attack in Kushar in Hajja province, killed 10 women and 12 children.
Another 30 people, including 14 under the age of 18, were wounded in the northern district, which is home to 100,000 people, was shelled.
Lise Grande, a UN Coordinator based in Yemen, said: “Many of the injured children have been sent to hospitals in Abs district and in Sanaa for treatment and several require possible evacuation to survive.”
Rights groups and the UN have criticised the Saudi-led and Western-backed coalition for air strikes that have often hit civilians, although the alliance denies doing so intentionally.
A coalition spokesman did not immediately respond to a request for a comment. Saudi-owned Al Arabiya TV accused the Houthis of responsibility. The Houthis are not known to have any aerial power.
The UN warned that thousands of civilians are trapped in Hajja province, as fierce clashes between warring factions intensify.
More than 5,300 families have been forced to flee Kushar and its surrounding area the last few weeks alone.
According to the UN, clashes broke out between the Houthis and the Hajour tribesmen in late January and fighting intensified in March with the start of air strikes there.
Hajjah's mountainous district of Kushar, 30 miles from the border with Saudi Arabia, has been hit particularly hard. Roads and all communication lines are cut and "thousands of civilians are reportedly trapped between conflicting parties", the UN said. The Houthis say their revolution is against corruption and accuse the Hajour tribesmen of accepting arms and support from Saudi Arabia.
Although both sides reached a cease-fire in the Arab nation’s main Hodeidah port in December, the Houthis still control most of the north and west.
"It is outrageous that innocent civilians continue to die needlessly in a conflict that should, and can be solved," Ms Grande added.
The ongoing conflict in Yemen has raged since 2015 when Saudi Arabia formed a coalition to fight back against Houthi rebels who took control of the capital and forced President Abdrabbuh Mansour Hadi to flee.
Tens of thousands of people have been killed and more than two million displaced in the impoverished country.
Additional reporting by agencies
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