The mother and father of Cheng Chieh, aged 21, were mobbed by reporters and some relatives of those killed or injured when they met the public for the first time since the bloodshed on Wednesday last week.
Wearing hats and face masks, the couple prostrated themselves three times at the subway train station outside Taipei where the college student was arrested.
"Although he is our child, the crime he committed is unforgivable," Cheng's sobbing father told reporters.
"I think he should be sentenced to death... he should face it himself. Only by so doing may the pains inflicted on the victims and the wounded and their families be slightly eased."
The father urged judges to pass sentence as soon as possible. "We hope Cheng Chieh can act in a correct manner during his next life," he added.
The families of some of the dead and injured said they would accept the couple's apologies while others termed them insincere, according to TV reports.
Despite protests from rights groups, Taiwan last month executed five death row inmates, nearly a year after six prisoners were put to death.
Cheng killed four people and wounded nearly two dozen others in the first fatal attack on the city's subway system since it began operating in 1996.
A spokesperson for the Panchiao district court said the suspect had admitted his guilt but declined to reveal further details.
Chen Kuo-en, chief of the New Taipei City police bureau, told reporters hours after the killings: "He told the police that he had wanted to do a 'big thing' such as this since his childhood".
"He said he had planned to do this after his graduation, but last week decided to act ahead of schedule."
The attacker had no record of mental disorder.
Local media said Cheng had been obsessed with online killing games and had written horror stories since high school.
Security has been strengthened in the metro, which transports around 1.85 million people per day.