Rohan Rhodes should have had blood tests on three occasions in the hours before he died at St Michael's Hospital in Bristol, coroner Maria Voisin said.
A narrative conclusion was recorded at Flax Bourton Coroner's Court.
The hospital trust said a safety check system has since been introduced.
Ms Voisin told an inquest into his death that there were "lost opportunities" to give Rohan earlier medical care before his death when the tests were not carried out.
Rohan, from Narberth, was born 15 weeks early at Singleton Hospital in Swansea in August 2012, but was transferred five weeks later to St Michael's Hospital in Bristol for specialist surgery to his heart.
The "extremely premature baby" died there aged just 36 days old.
The inquest into Rohan's death has heard how he deteriorated soon after a ventilator was removed without the consent of his parents or senior hospital staff.
Medical staff wept as they told the three-day hearing how the baby's condition dramatically worsened.
The inquest heard the baby was not allowed to die in his mother's arms despite her wishes because staff were unable to remove lines from his body.
Rohan had been transferred to St Michael's Hospital for surgery to repair an open heart duct.
His parents Alex and Bronwyn Rhodes told the coroners' court on Monday about concerns they had about their son's treatment.
They said that Rohan's feeding tubes were inserted "aggressively" and caused him pain.
The couple said they did not want any other child to suffer as Rohan did.
Following the inquest Mr and Mrs Rhodes said they were planning to take legal action against the University Hospitals Bristol NHS Foundation Trust, which runs the hospital where their son died.
Bryony Strachan, of the University Hospitals Bristol NHS Foundation Trust, apologised that opportunities to carry out tests on Rohan were missed, and said safety systems have since been introduced.
"The coroner has confirmed that what those results would have been remains unknown, but we are very sorry that those three checks were missed, within continuous monitoring of Rohan's critical condition within NICU (neonatal intensive-care unit)," she said.
"We have already put in place clear requirements for blood gas measurements in babies on respiratory support and have implemented a system of safety checking shared by both medical and nursing staff looking after individual babies," she added.
On Tuesday the court heard from medical staff at the hospital who said the plan had been to keep Rohan on the ventilator.
However, the hearing was told advanced neonatal nurse Amanda Dallorzo took the "autonomous" decision to extubate (remove a tube from the airway), remove the baby from the machine, and apply a breathing mask instead.
Rohan's condition dramatically deteriorated and he died the following day.
Mrs Rhodes said: "We were terrified and were watching all this in horror. He was so pale and lifeless during this time, we were terrified we were losing him."
Dr David Harding, Rohan's consultant, told the inquest that the weekend Rohan died was the "busiest and worst weekend of his career".
The court also heard that there were fewer nurses on the ward that weekend than there should have been.
A "root cause analysis" report looking at the cause of Rohan's death concluded the ward was understaffed.
The parents of a severely premature baby from Pembrokeshire say they will take legal action against the health trust in charge of the hospital where he died.