Mohammed Abdel Salam, a Houthi spokesman, identified the pilot as Lieutenant Shamsan Mohammed Abdu Meflih.
'Scorched Earth' campaign
Troops continue to press their seven-week-old offensive against the Houthis in the northern mountains, with no sign of the conflict ending.
Five fighters and four soldiers are reported to have been killed in fighting in the Harf Sufyan district of Amran province, on the road linking the capital Sanaa with Saada, the centre of the region of the same name which borders Saudi Arabia.
On Wednesday, 28 fighters were also killed in clashes near Saada.
The army launched operation "Scorched Earth" on August 11 in an attempt to finally crush an uprising in which thousands of people have been killed since it first broke out in 2004.
The United Nations estimates that 55,000 people have fled their homes because of the conflict.
The authorities accuse the Houthi fighters of seeking to restore the Zaidi Shia imamate that was overthrown in a republican coup in 1962, triggering an eight-year civil war.
The fighters deny the charge and say they are fighting to defend their community against government aggression and marginalisation.
A minority in mainly Sunni Yemen, the Zaidis are the majority community in the north.