The plot, at Bridlington Cemetery on Sewerby Road which had been visited by the family for more than 20 years, was dug up by mistake and then used to inter another body on Friday.
The family said it was dismayed, angered and horrified by the situation.
East Riding of Yorkshire Council has apologised unreservedly to the family for any distress caused by the mistake.
Andrew Lowe said: “Last Wednesday my sister Maureen went down to the cemetery to place some flowers on our mum and dad’s grave.
“When she got there the person who digs the graves was removing the headstone and Maureen said to him ‘what are you doing’.
“He said that he was removing the headstone so it doesn’t get damaged while he dug a grave. She assumed that the plot behind or to the side was being dug up ready for a funeral.
“She put the flowers down and left.
“On Monday morning (November 15) she returned to the area to find out that they had dug my mum and dad’s grave up last Wednesday and buried someone on top of my mum on Friday.
“Maureen got in touch with the grave digger who went to the office and made numerous phone calls. After about an hour he came to the grave and said ‘I don’t know what has happened’.
“He informed us that council officials were coming down to look at the site tomorrow (Wednesday, November 17).
“However, one of our friends said that there was some activity at my mum and dad’s grave.
“There were people there in suits, the grave digger and an apprentice, stacks of wooden boards and a digger.
“The head of the council department then rang my sister saying they have sorted the issue, dug the right grave which was the one behind, and moved the body.
“The council had made a huge error, they actually buried someone in the wrong plot.
“They have apologised, which we accepted. However, they said that the body was slid across into the new grave but the area between looks like it hasn’t been disturbed so clearly it has been lifted out of our mum and dad’s site and into the new plot.
“My dad was buried 24 years’ ago and my mum 21 years’ ago and this horrific mistake will have caused upset to two families.”
A spokesman for East Riding of Yorkshire Council said: “We apologise unreservedly for this genuine mistake.
“We contacted the affected families as soon as this was discovered and we explained the situation fully and have apologised sincerely.
“We have now rectified this situation in a sensitive manner.
“We are investigating this situation and we acknowledge this should never have happened.”