In one case, the parents said an early indication of Down Syndrome was detected by ultrasound in the first trimester and a further test was recommended, but doctors who were consulted next did not confirm.
"Had the presence of Down Syndrome been diagnosed at the time of the first trimester ultrasound and/or at the time of the second trimester ultrasound, a time frame which permitted the termination of the pregnancy, then the (mother) would have terminated the pregnancy," the statement of claim says.
The girl, now aged four years, attends a specialist kindergarten, was born with heart, kidney and thyroid problems, can't walk, and needs help feeding, her father said.
"Don't get us wrong: we love our daughter. She's part of our family, and we treat her like gold," he said. The issue is the failure of responsibility, he argues. The Royal Women's Hospital, targeted by the suit, is yet to file a defence.
In the second case, the parents allege a failure to "exercise reasonable care" of the mother, "advise (her) in relation to the risks of Down syndrome given her age" and "provide (her) with the option of ante-natal screening.
"In the event that Down Syndrome had been detected, (she) would have elected to undergo a termination of her pregnancy," their statement of claim says.
The parents claim to have suffered "depression, shock and anxiety" and "pain and suffering during pregnancy and delivery."
But the hospital authorities concerned deny any wrongdoing and say the family has failed to show it suffered a "significant injury" as defined by law.
If there is no out-of-court settlement, civil trials will start, it has been reported.