In Elizabeth George Speare’s The Bronze Bow, Daniel’s father was crucified by the Romans, and his mother died of grief and illness a few weeks later.

Daniel was only eight years old at the time, but he remembers every bit clearly, and he tells the story to Joel and Thacia. It all started when Daniel’s uncle, his father’s younger brother, did not have enough money to pay the tax collector. The Roman soldiers arrested him when he tried to make a fuss, and although Daniel’s father raised the money to pay the tax, the Romans decided that they would make Daniel’s uncle work in the quarries instead.

Daniel’s father and some others armed themselves and tried to rescue Daniel’s uncle, but things went very wrong. A soldier died, and Daniel’s father and his companions were all captured and crucified.

Daniel’s mother would not leave the crosses for two days until her husband died. It was cold and wet, and she developed a cough. This, along with her grief, took her life quickly. Daniel and his younger sister, Leah, went to live with their grandmother. But Daniel had made a vow to hate and kill the Romans.

How do Daniel's parents die in The Bronze Bow?

Just a bit to add to tthaker's correct answer.  Daniel's mother died not only from grief, but also from exposure, probably from pneumonia.  The deaths of Daniel's parents have additional health ramifications.  Daniel's sister, Leah,  who found her mother at the foot of her father's cross, is so traumatized that she becomes agoraphobic (fear of going out of one's home) and suffers other fear-induced disabilities.  Although modern readers would probably immediately recognize the trauma to the young girl, the explanation in Daniel's time was that the girl was possessed by demons.

Daniel's aging grandmother also suffers physically because of his parent's death.  Forced to work even harder in the fields, gathering the grains left behind by the threshers, she eventually wears her body completely out.  Her death is the reason that Daniel returns from the mountain, where he has been holed up with Rosh, to be with her on her deathbed.   

How did Daniel's mother die in The Bronze Bow?

Daniel’s mother died of grief and from contracting an illness after the crucifixion and subsequent death of her husband. Exposure to the cold weather when mourning her husband may have caused the terrible illness, perhaps pneumonia, that led to her death. She stayed at the site of the crucifixion for a few days until her death.

Daniel’s uncle failed to pay his taxes, and his punishment was to work in the quarries. However, Daniel’s father tried to intervene. Together with his friends, he attempted to free the uncle as the prisoners headed for the quarries. The rescue mission was botched, and they were all arrested. The punishment for the act was crucifixion. Daniel and his sister were thus orphaned at a tender age. They were taken in by their aging grandmother.