Pinki, 29, has been suffering from cerebral palsy -- a disorder that causes physical disability -- since she was four years old.
The parents say they can no longer pay for the medical expenses nor can see their daughter in pain.
"We cannot watch her in so much pain anymore. Please allow her mercy killing or provide her medical help because we cannot afford it any longer as we are poor," Pinki's father Kishori Sao, a resident of Mor village in Patna district, said here Friday.
"We have already spent whatever we had for her treatment, but she not showing any sign of improvement," the father added.
Sao says he has no option other than seeking the mercy killing of her daughter.
Pinki's mother Rukmani Devi said her daughter is suffering from the rare disease for the last 25 years.
"We have consulted doctors, ayurvedic practitioners, quacks as well as priests for her treatment but failed. Even visiting temples, churches and gurdawaras has yielded no results," she said.
She said some doctors here advised them to take their daughter to a reputed hospital in Delhi.
"But where is the money. We are fighting for survival, how can we take her to a big hospital. Urging the president to permit mercy killing is the only option left with us," the mother said in a choked voice.
"We are now begging for her peaceful death," she said.
Doctors of the Mokama Referral Hospital, where Pinki was treated, said she is 90 percent disabled due to the disorder.
Her condition can be improved by treatment through physiotherapy, the doctors said.
According to medical experts, the disorder often occurs at a premature age -- during pregnancy, during childbirth, or after birth up to about age three -- affecting the brain development of children. About two percent of all cerebral palsy cases are believed to be due to a genetic cause.