Two civilians were also killed and 16 wounded when rebels fired Katyusha rockets into a residential area of Taez, they said
The clashes have eliminated hope of the warring parties abiding by a truce announced by US Secretary of State John Kerry that was to have taken effect on Thursday.
Loyalist military sources told AFP that 13 rebels and eight pro-government forces were killed over the past 24 hours, while the rebels reported dozens of casualties in shelling of a local market.
Two civilians were also killed and 16 wounded when rebels fired Katyusha rockets into a residential area of Taez, they said.
For its part, rebel-controlled sabanews.net website reported that pro-government forces fired artillery rounds into areas east of the city.
A rocket hit a market selling the mild narcotic leaf qat, popular among Yemenis, killing 24 people and wounding 27 on Thursday, sabanews.net said.
It cited the health chief of Taez province, Watheq Faqih, as saying that the toll could rise as rescue workers had not been able to gather the body parts of all the dead because of artillery fire on Friday.
The medical charity Doctors Without Borders (MSF) said emergency rooms it supports or manages in Taez province had received a total of 21 dead and 76 wounded.
Among those killed was a watchman who worked at the MSF trauma centre in Taez.
He was killed while "off duty when a blast hit a local market in the neighbourhood", said Djoen Besselink, who heads the MSF mission in Yemen.
"Fighting has been intensive in Taez during recent days, and hospitals on both sides of the front line have received a continuous influx of war-wounded civilians and fighters," MSF said.
Pro-government forces are pressing on with a four-day-old offensive to recapture the presidential residence and police headquarters in the southwestern city, while the Shiite Huthi rebels have brought in reinforcements.
The fighting continued despite Kerry's announcement of a new ceasefire.
The US chief diplomat said on Tuesday that rebels were ready to observe a ceasefire plan taking effect from November 17, but President Abedrabbo Mansour Hadi's government said it was not aware of any new peace initiative.
The Huthis and the party of their ally, former president Ali Abdullah Saleh, confirmed their commitment to the truce, in a statement on the rebel television channel Almasirah's website.
A Saudi-led coalition which backs Hadi's internationally recognised government told AFP Thursday that "until now there is no demand from the legitimate government (of Yemen) to observe a ceasefire", adding that operations will continue.
The United Nations says more than 7,000 people have been killed and nearly 37,000 wounded in Yemen since the Arab coalition began a military campaign in March 2015.