Heavily pregnant Cheryl, 35, from Essex, was days away from giving birth when a scan showed there was a problem with blood supply to her baby from her placenta.
The couple were told that - despite the scan's findings - there was not enough staff and equipment to deliver their baby, and they were sent home.
Cheryl then had to return to Southend Hospital in Essex to deliver her stillborn son, who the couple named Hayden. Cheryl's labour lasted 16 hours and the little boy was born weighing 6lb, 8oz.
The hospital has to cut £130million from its books by 2014.
Cheryl was due to give birth on June 25, and the problem was discovered on the previous Thursday. She was monitored all weekend until 12pm on Sunday, when she was told her baby's heartbeat could not be found.
"I think what has happened is as a direct result of the budget cuts and lack of funding," says 40-year-old Alex. "The NHS is supposed to be a 24-hour service. If they had delivered him during the weekend, I am sure that he would still be alive and here with us.
"We literally begged the doctors to keep Cheryl in during the weekend and deliver our baby but they flatly refused. The consultants were telling us that they could not deliver him because of a lack of staff and equipment and they do not induce people during the weekend.
"If they had delivered him during the weekend, I am sure that he would still be alive and here with us. We are both devastated and very angry about the way it was handled.
"It was on Cheryl's notes from her consultant saying 'any problems, deliver straight away'. Cheryl was just three days from full term so it was not a problem for him to be delivered there and then.
"I have been on the phone organising Hayden's funeral and which hymns will be sung - I should not be doing that, it should be looking forward to his Christening."
Sue Hardy, Southend Hospital's director of nursing, said an investigation had been launched: "We are extremely sorry to hear of the sad loss of Cheryl and Alex's baby. We would welcome the chance to meet with the parents to explain to them the form the investigation will take."