The clashes, which erupted late yesterday near Sanaa's Saleh mosque, killed nine Huthis and five supporters of former president Ali Abdullah Saleh, medical sources said.
The infighting threatens to unravel the fragile rebel alliance that has been battling the Saudi-backed government of President Abedrabbo Mansour Hadi in Yemen's war.
Late-night mediation attempts betwen Saleh and the rebels, who jointly control Sanaa, failed to reconcile both sides, sources in Saleh's General People's Congress political party said.
Rebel chief Abdulmalik al-Huthi had on Wednesday urged supporters across the country to head to Sanaa for the event.
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Saleh and the Iran-backed Huthis, also known as Ansar Allah, have accused one another of inciting Wednesday's unrest.
"The General People's Congress and its allies hold Ansar Allah fully responsible for every drop of blood shed among the Yemenis... and warn against all acts that, rather than serve national unity, threaten our internal unity and cohesion," the party said in a statement.
The rebels' interior ministry blamed forces loyal to Saleh for the clashes in a statement released late yesterday.
"We were surprised when these armed forces inside the mosque opened fire on police without warning, which forced police to fire back," it said.
The rift between Saleh and the Huthis goes back months, with the former president slamming the Huthis as "militias" and the rebels threatening Saleh loyalists after armed violence left two dead in Sanaa in August.
The Huthis have also accused the former president of accepting funds from the Saudi-backed Hadi government.
The United Nations has warned that the country faces mass famine unless the Saudi-led coalition allows more food aid to enter the impoverished country.
Saleh ruled Yemen from its unification in 1990 until he resigned under pressure in 2012, ceding power to his then vice-president Hadi.
He fought six wars against the Huthis when he was president, but joined forces with them to take over the capital in 2014.