Sarae Thompson-Haynes was born on February 11 but by March 14 she had developed a bad cough. Her concerned mother Chelsea Thompson, 21, took her to the GP to have her checked over. By evening, Sarae's condition had worsened and she took her back to the out of hours clinic.
Chelsea told her local newspaper the doctor gave them an inhaler.
By the next day Sarae's condition had deteriorated and after calling NHS Direct, Chelsea and her partner Todd Haynes, 23, took their daughter to Ipswich Hospital's A&E department where she was given chest X-rays and a series of tests.
"Doctors at Ipswich mentioned she might have bronchitis. We felt like we always had to ask doctors what was going on. Communication was terrible.
Miss Thompson added: "It wasn't until she was transferred to Addenbrooke's that a test for whooping cough was done.
She was eventually taken to Great Ormond Street Hospital and died before a test for the illness, carried out at Addenbrooke's Hospital in Cambridge, showed positive for whooping cough.
Her parents said they would lodge a formal complaint against their GP surgery in Harmoni, Ipswich, which runs the out-of-hours service, and Ipswich Hospital.
They said they felt they had not been taken seriously before their baby's death in March.
"We are angry and devastated at the same time. Sarae was absolutely beautiful. It is something you don't wish on anyone," said Chelsea.
The cause of death was recorded as cardiac failure, severe sepsis and pertussis pneumonia, known as whooping cough.
Chelsea added: "I can't say it would have changed the outcome for our little girl but I wish people had listened to us when we knew there was something wrong. I was a first-time mum and I just felt I wasn't being taken seriously.
"It is rare but doctors can test for it, you don't have to end up in hospital."
Cases of whooping cough are currently at a 20-year high, and five babies have already died from the infection.
A spokesman for Hawthorn Drive GP surgery said it could not discuss the details of the case. The practice manager said: "The whole team at the surgery offer their sympathies to the parents and family."