A number of witnesses reported seeing a 21-year-old man from the Keighley area, allegedly punch the couple on Duncombe Place at around 7.15pm on Saturday 30 July 2012.
The man was arrested at the scene, however the victims had left the area before officers arrived.
Despite efforts to trace the elderly couple they remain unidentified and officers are appealing to them to come forward.
It is believed that the elderly man was wearing a navy jumper over a pink shirt and was celebrating his 74th birthday.
PC Nicola Elliot, of York police, said: “This must have been an upsetting experience for the elderly couple and, understandably, they may have wanted to get away from the scene as quickly as possible.
“It is important, however, that we trace them so they can help us with the investigation.
“It is not believed that either of the couple were seriously hurt in the incident but I am keen to check on their welfare and get their version of events.
“If you were the elderly couple involved or know who they might be, please contact the police or Crimestoppers as soon as possible.
“I would also appeal to anyone else who witnessed the incident to come forward.”
Anyone who can help the investigation is urged to contact North Yorkshire Police on 101 – select option 2 – and ask for Nicola Elliot or York police.
Alternatively, Crimestoppers can be contacted anonymously on 0800 555 111.
Please quote reference number 12120105623 when passing information about this incident.
* The 21-year-old man who was arrested in connection with the incident has been released on police bail as enquiries continue.