Members of the battling Toori and Mangal tribes are using heavy weapons, including rocket-propelled grenades and mortar guns, witnesses said.
The government on Friday warned the warring tribesmen to halt the fighting by Monday or a military offensive would be launched to restore order.
"In today's and yesterday's clashes at least 23 people have been killed on both sides and 28 others were injured," a security official said on Sunday.
The clashes erupted last week when a tractor belonging to a Shia Toori tribesman was set on fire.
Sectarian violence gripped Kurram in April 2007 and most parts of the district have been cut off from the rest of the country through land routes, triggering a severe shortage of food and medical supplies.
Four children died on Saturday due to unavailability of medicine in a hospital in Parachinar, the main town of Kurram, according to the Associated Press of Pakistan news agency.
The road linking Parachinar with the neighbouring North-West Frontier Province was closed 10 months ago, leaving the town's population of nearly 500,000 in a virtual state of siege.