Hayley Clarke and Peter Green, from Stowmarket, Suffolk, said they raised concerns over little Harry's weight but were told not to worry.
The true severity of the baby's condition only came to light when he was two months old and was referred to hospital.
Doctors found that Harry had a heart defect that had caused a narrowing of his aorta.
Within 24 hours, Harry underwent a three-and-a-half hour heart operation at the Evelina Hospital in London.
Now Hayley, 33, and Peter, 52, have lodged formal complaints with Suffolk County Council, which is responsible for health visitors, and Stowhealth, the GP practice where Harry was assessed.
The couple noticed Harry was struggling to feed and was not putting weight on soon after he was born on November 13 last year.
At weekly weigh-ins with health visitors, the couple repeated their concerns – but they say they were told their son was fine.
A Suffolk County Council spokeswoman told the local newspaper: "We work extremely hard to make sure that all the necessary steps are in place to support young people and their families to be healthy and happy.
"Where it is highlighted that this may not be the case we take the steps required to ensure we learn and improve the service we offer.
"Suffolk's Health Visitors provide a vital service to all families and young children across the county with the primary focus of ensuring parents are supported and children are well cared for."
Dr Baber Yusaf, senior partner StowHealth said: "We sympathise with the family at this difficult time. Our professional duty of confidentiality relating to the care of our patients means I cannot comment on this case. The whole team at StowHealth always strive to do the best we can."