British-born David Cushworth and Salvadorian-born Mercedes Casanellas accused a doctor at the hospital of swapping their child as part of a suspected trafficking plot.
DNA tests then confirmed the baby they took home with them was not their child.
Mrs Casanellas first raised concerns when the baby's skin seemed darker than its mother and fathers.
Speaking in the Metro, Mr Cushworth told reporters the couple had "no idea" before now that they had taken home the wrong baby.
They wrote on Facebook: "There are no words to express what our heart feels to have our baby at home! Thanks to all who joined our pain and fed our hope.
"Thank God our family has gathered, for the moment we are not allowed to upload photos but soon we'll give more details of the outcome of this miracle."
Gynaecologist, Dr Alejandro Guidos, who is now under investigation, denies the charges.
The parents accuse Dr Guidos of planning swap their lighter-skinned son with a darker-skinned infant to sell on to traffickers.
Dr Guidos says this was an innocent mix-up.
Mr Cushworth, originally from Bradford, and Mrs Casanellas who now live in Texas, have decided to name their son Moses.
Speaking to the Mail Online, Mr Cushworth's mother, Lynne, said: "Moses was lost for three months and now he's been found."
Mr Cushworth said: "This is an answer to so many prayers.
"All the circumstantial evidence pointed to a criminal act, but the official story we are being told appears to show it was a mistake, a mix-up.
"I feel for the other family. We know nothing about them, except that they only just found out today that their baby isn't really theirs. It must have been devastating."
