"Denial of access to travel has condemned thousands of Yemenis with survivable illnesses to death", the NRC's Director in Yemen Mutasim Hamdan said in a statement.
Aid groups have issued a statement calling on Yemen's warring parties to reopen the country's main airport, saying the year-long closure has trapped thousands of sick patients and is stopping vital humanitarian supplies.
Yemen's foreign ministry accused the rebel militias of attacking the Sudanese embassy in the Yemeni capital Sana'a which is controlled by the Iran-backed alliance of Houthis Ansarullah group and former President Ali Abdullah Saleh's General People's Congress (GPC).
Since the escalation of conflict in Yemen in March 2015, violent attacks have resulted in more than 54,000 casualties, and devastated existing infrastructure.
Now, the number of people needing life-saving healthcare is around 20,000 over the past two years because of the violence, the group added.
According to the World Health Organization, Yemen is now coping with the "world's worst cholera outbreak", which has claimed the lives of at least 1,800 people and infected more than 370,000 others. Last month, a revised United Nations humanitarian assessment said the number of people in need of assistance had risen from 18.8 million to 20.7 million, a figure equivalent to nearly three-quarters of the total population. "Thousands of women, men and children who could have been saved lost their lives".
The closure has left many Yemenis with no safe means of transport in or out of the country, the NRC added in a statement. Mohammed's father died less than a day before his flight.
Restrictions imposed on Yemen's airspace by the Saudi-led coalition resulted in the official closure of Sanaa airport to commercial flights on August 9, 2016.
The aid groups said: "The current cholera outbreak and near-famine conditions in many parts of Yemen make the situation far worse".
Yemen's health ministry estimates that 10,000 Yemenis have died from critical health conditions for which they were seeking global medical treatment, the statement said, adding it was unable to verify the figure.
Riyadh and its allies have been blocking aid delivery to Yemen.
Prior to the escalation of conflict in Yemen, an estimated 7,000 Yemenis were travelling overseas for medical treatment not available in the country, a number that has grown exponentially since March 2015, when a Saudi-led military coalition began airstrikes against Iran-backed Houthi rebels who orchestrated a bloody coup against the Hadi government in September 2014.