The baby boy, known only as Donald, was born prematurely two months ago and the cause of his death is unknown.
Donald ended up sleeping in the car after his parents were evicted from a house they were renting and couldn't afford the deposit for a new place.
Jane gave birth to Donald, her first child, on July 6 by emergency caesarean after suffering from pre-eclampsia and kidney problems.
She said at the time of his birth: "I love him to bits. I just want him to be safe.
"He's my baby. I'm not going to let any harm come to him. We're all together, we're still a family unit."Claire Matthews, who runs the Hope for Food soup kitchen in Bournemouth, worked closely with the family.
She said the family are "grief-stricken".
"They are in a very dark place at the moment, they have nothing to live for now," she said.Claire said Donald's mum, known only as Jane, had moved to Dorset from Kent while still pregnant to escape the threat of domestic violence from an ex-partner.
Hope for Food has now offered to help pay for tragic Donald's funeral.
It is believed his devastated parents are currently living in a bed and breakfast and have said in the past they have been let down by local councils unable to find them permanent housing.
Claire said: "Because the mother was from Kent the local authorities wouldn't help her, despite the father being from Bournemouth.
"They were never after hand-outs - they just wanted support.
"After Donald's birth they moved back to Kent because that's what they were told they needed to do in order to get the help they needed.
"The authorities there put them up in a B&B but then they phoned to say Donald had died. We don't know why at the moment, but there will be an inquest to find out.
"It is disgusting that something like this could happen in this day and age. If they had just had some help from the local authorities when they first asked then Donald would still be here."Ben Tomlin, housing services manager for the Borough of Poole, said: "Borough of Poole was contacted by the mother and though she was not eligible, she was provided with as much support and guidance as possible.
"She last approached us in May and hadn't returned to Poole for help after her baby was born. Our deepest sympathies are with her at this time."