The staff of a children's hospital in the Shaar district were forced to take shelter in the basement.
Activists say at least 32 people, including children, have been killed in Aleppo over the past two days.
The air strikes resumed on Tuesday after a three-week moratorium declared by the government's ally Russia ended.
Activists confirmed the resumption of government air strikes, amid reports by state media of large troop deployments on several fronts ahead of a major ground assault.
The UK-based Syrian Observatory, a monitoring group, said missiles fired by jets, barrel bombs dropped from helicopters and artillery targeted several areas of eastern Aleppo on Wednesday, including Shaar, Sukkari, Sakhour and Karam al-Beik.
At least 21 people, including five children and an emergency worker, were killed, the group said.
The Independent Doctors Association reported that a children's hospital it supported had been badly damaged.
It quoted the hospital's director, Dr Hatem, as saying he was trapped in the basement. "The planes are up above. We can't get out. Maybe we can protect ourselves in this room," he added.
Rescue workers from the Syria Civil Defence, who are also known as the White Helmets, meanwhile confirmed a paramedic had been killed in Karam al-Beik.
"The helicopters won't stop for a single moment," spokesman Bebars Mishal told the Reuters news agency. "Right now, the bombing won't let up.