City of York Council social workers were accused of failing to properly review the case of the baby boy, whose parents were placed under strict supervision arrangements after concerns were raised for his welfare.
The youngster was found to have rib injuries when he was taken to hospital with breathing difficulties and his parents were barred from spending time with him alone, the Local Government and Social Care Ombudsman said.
The mother and father told social workers and a doctor that his injuries could have been from previous hospital treatment, but they were prevented from seeing him without supervision and on one occasion were unable to see him at all.
To add to their torment, the measures also meant their other two children had to stay with their grandparents.
The boy, whose identity has not been released, died nine weeks after his admission to hospital and the council later decided the injuries could not have been "attributed to parental care".
Ombudsman Michael King said: "Nobody could criticise the social workers in this case for starting the safeguarding action.
"But what they should have done was keep the situation under review, especially once it became clear there was very little risk to the child, and his condition deteriorated.
"This would have been a horrifically stressful time for the family, at a time when their world must have felt like it was falling apart.
"Social workers should have done more to facilitate the parents' visits, for example by contacting social workers in the neighbouring authority, or the hospital's own patient liaison services."
The ombudsman's probe found that the council failed to visit the baby in hospital and the care plan did not consider the baby's "emotional needs".
The baby was moved to another hospital in a nearby city and there was "no evidence" that the council looked at alternative solutions to help the parents visit their child.
Despite the baby's condition deteriorating, the supervision restrictions were not changed, while it took a month after the boy's death for the couple's children to be allowed home.
The ombudsman said a court had criticised the council's handling of the case.
The council has also come under fire for taking nearly a year to respond to the family's subsequent complaint.
Instead of taking a maximum 65 days to respond to their complaint it took 343 days.
The Ombudsman said City of York has agreed to apologise to the family and pay them £2,000 for "the distress caused by its actions".
Maxine Squire, interim corporate director for children, education and communities at City of York Council, said: "We are extremely sorry for the distress caused and have apologised unreservedly to the family.
"We fully accept the Ombudsman's findings and recommendations. We have already taken action to ensure that lessons are learnt from this case and that our procedures are improved."