Dave Mears and Emma Collins had been allowed to take their son Logan home from Pontefract General Infirmary less than 24 hours after he was born, despite their concerns about his breathing.
A community midwife is said by the couple to have found nothing wrong with Logan during a visit to their home in Woodlesford, Leeds, the day before he died.
In fact,the baby had a pneumococcal infection which eventually caused his kidneys to shut down and led to the onset of septicaemia.
He had caught the infection from his mother, who had herself been feeling ill in the days before and after the birth on July 28 this year.
The couple say a string of medics failed to recognise the seriousness of her condition, with one doctor allegedly telling her she was suffering from back muscle strain.
Staff at Wakefield's Pinderfields Hospital quickly realised she was sick when they saw her on the night Logan died.
He had been taken to Pinderfields by ambulance from the couple's home while Emma, 19, was driven to hospital by a relative.
Tests revealed she was suffering from severe pneumonia, septicaemia, heart inflammation and lung abscesses.
Emma spent 33 days in intensive care at Pinderfields but since September has been slowly recovering at the house she and Dave share with her parents.
The couple are now suing for clinical negligence, claiming that staff at Pontefract failed to spot both Emma and Logan were seriously ill.
Reliving the night of Logan's death, Mr Mears, 28, said: "He was asleep in a Moses basket in our room and Emma was in bed by the side of him, although she was feeling so ill she didn't really know what was going on.
"I'd been checking on him every 20 minutes and he seemed fine.
"Emma's parents had been out and got in about twenty past 10. Her mum went to check on Logan and he'd stopped breathing."
A spokesman for Leeds Teaching Hospitals NHS Trust said: "We believe our community midwife acted appropriately but we will of course be happy to do our best to answer any questions the family has."
Tracey McErlain-Burns, chief nurse at the Mid Yorkshire Hospitals NHS Trust which runs Pontefract Infirmary, said: "Because Ms Collins was admitted to intensive care (at Pinderfields], we carried out a review of her and her baby's care whilst at Pontefract.
"We would welcome the chance to meet with Ms Collins to talk through our findings."
NHS Leeds, the body that co-ordinates healthcare in the city, said an investigation had been launched into the case.
A spokeswoman for Oulton Medical Centre said: "We're working closely with our colleagues at NHS Leeds to fully understand all aspects of the care given."
Pneumococcal infections can spread in the same way as a cold or the flu.