The latest attacks come after U.N.-led peace talks reached a dead end last week, with fierce clashes reported throughout the country.
Saturday's incident has also renewed concerns over the mounting number of civilians -- many of them children -- who have died in the conflict.
However, the coalition countered that the weekend airstrikes targeted a "major training camp for the militia" -- and resulted in the death of a Houthi rebel leader. It said Houthi recruitment of kids in the conflict was to blame for the fatalities.
"The aircraft has bombed a training camp for the coup militias called (Huda) in Saada," coalition spokesman Major Ahmed Asseri told CNN in a statement Sunday.
He did not specify which school of the two attacked on Saturday he was referring to.
It "confirms the Houthis practice of recruit and subjecting children to terror," Asseri continued.
According to Asseri, Houthis regularly recruit children "and use them as scouts, guards, messengers and fighters," ultimately "subjecting them to injury and murder."
The coalition -- made up of several Arab countries -- began a military campaign in Yemen in March 2015 aimed at preventing Houthi rebels allied to Iran and forces loyal to Yemen's deposed President Ali Abdullah Saleh from taking power.
'No way those were fighters'
However, UNICEF's representative in Yemen told CNN the children were too young to be fighters.
"We've had a verification team who went to the site and was there on the day. We've been to the hospital and we've spoken to parents. Many of these children were six-years old, eight-years old. There's just no way that those were fighters," UNICEF's Julien Harneis said Sunday.
Ten children died and 28 others were injured in the attack on a private school in the Haydan district of Saada province, which occurred when children were leaving the school and were on their way home.
The children were between the ages of 8 and 15, according to Medecins Sans Frontiers' official Twitter account.
Another four children and three adults were killed in a coalition airstrike on a school in nearby Razeh district.
'Another 10,000 children will die'
Since the failed peace talks, Harneis said that the country has seen a spike in violence, paralyzing the country's infrastructure.
"The airspace into the airport has been closed for the last six days. No civil or humanitarian flights have been able to get in or out. The health system is on the verge of collapse. All across the board we have a terrible situation in Yemen," he said.
"Today, the money for health centers to run across the country is no longer available. When we try and give money to our partners across the country, they can't take money out from the bank. That means health services for vaccination, for measles, for polio, for routine vaccinations, for treatment for things like malnutrition, it's beginning to unravel across the country," Harneis added.
"We estimated that at the very least another further 10,000 children will die of completely preventable diseases that would not have happened were it not for this conflict."
Condemnation and call for investigation
United Nations Secretary-General Ban Ki Moon condemned the attack Monday and called for a swift investigation, his spokesperson said.
"The Secretary-General notes with dismay that civilians, including children, continue to bear the brunt of increased fighting and military operations in Yemen," a statement from the Secretary-General's office read.
"He is urging all parties involved in the conflict to renew their engagement with his Special Envoy for Yemen to find a negotiated solution," it said.
But the U.N. itself came in for criticism.
"The attack was on a school and that's common and not the first time children and schools were attacked. The massacres against civilians are happening while the U.N. watches silently ... not stopping the crimes in Yemen," Mohammed AbdulSalam, a spokesperson for the Houthi rebels, told CNN Monday.
"[The] U.N. has betrayed the children of Yemen with its silence towards these Saudi crimes."
According to U.N. figures, the Saudi-led coalition was responsible for 60% of the 1,953 children recorded as killed or maimed in the conflict in 2015.
Medecins Sans Frontiers and UNICEF also objected to the attacks.
Scandal over U.N. blacklist
In June, Ban Ki-moon revealed he received "undue pressure" from the coalition to remove it from a blacklist of groups violating children's rights in armed conflict.
A U.N. official told CNN that Saudi Arabia threatened to pull hundreds of millions of dollars in financial contributions to U.N. humanitarian agencies and causes if they were not taken off the blacklist.
The Saudi government has vigorously denied this. Saudi Arabia's U.N. ambassador Abdallah al-Mouallimi dismissed those allegations as "rumors" circulated by people who "try to damage our reputation."