Melvyn Marcel was appealing against the decision, made by a judge earlier this year, that burials should be organised for his mum Hilda and his dad Eugenios.
However, appeal judges at the Court of Session rejected his argument.
Eugenios and Hilda Marcel's bodies were found in a former fishmonger shop in 2002.
In February, councillors gained permission from Lord Mullholand to arrange a burial for the couple.
Mr Marcel wanted to build a fridge at his Edinburgh home then plans to build a private mausoleum in the grounds of his property.
He eventually planed to take his parents' bodies to be buried in the West Bank in the Middle East.
The council argued they had a statutory duty to dispose of the bodies.
On Friday, Mr Marcel told the court Lord Mulholland's decision was wrong.
He told judges Lord Carloway, Lord Brodie and Lady Clark of Calton: "I'm fighting for justice but nobody's helping me."
Mrs Marcel died in 1987 from lung cancer whilst her husband passed away from prostate cancer aged 91 in 1994.
Their bodies were embalmed and a relative regularly visited them at the shop in Polwarth.
Police discovered Mr and Mrs Marcel's bodies during an investigation into alleged fraud at a funeral home in West Lothian.
It was claimed that staff at the Broxburn undertakers had been paid to preserve the remains and four employees were sacked.
However, none of those who were allegedly involved in the fraud were charged or prosecuted for any offence.
Since then the Marcels' bodies have been stored at Edinburgh City Council's mortuary with officials unable to find them a more suitable resting place without the consent of the family.
Giving the appeal court's decision, Lord Carloway said that Lord Mulholland's decision was legally correct.
He added: "No valid defence has been established in this action.
"It is the court's decision to adhere to the Lord Ordinary's interlocutor."
The son of a couple whose bodies have spent almost 15 years in an Edinburgh mortuary has lost a legal bid to stop the council arranging their funerals.