Miss Buckley's disappearance is said to be completely out of character Police are continuing to search the park where a missing student's handbag was found.
Karen Buckley, 24, from Cork in Ireland, left a Glasgow nightclub in the early hours of Sunday morning and went with a man to a flat, where she spent several hours.
She has not been seen since, but her handbag was discovered in the city's Dawsholm Park on Tuesday afternoon.
Officers have said they are "gravely concerned" for her wellbeing.
Miss Buckley's parents have flown to Scotland from their home in Ireland.
Her mother, Marian, told a news conference on Tuesday: "We just want Karen home safely, we are desperate. She is our only daughter, we love her dearly.
"If anybody has any information please come forward, we would dearly appreciate it."
Miss Buckley, whose disappearance is said to be completely out of character, was last seen on CCTV talking to a man outside the Sanctuary nightclub in Dumbarton Road.
The pair travelled in the man's car to his flat in Dorchester Avenue, in the Kelvinside area of the city. It is said that she left the flat at about 04:00 on Sunday and was planning to walk back to her flat in Hill Street, close to the city centre.
The two addresses are about four miles apart.
Police have stressed that the man is helping them with their inquiries, but is not a suspect at this stage.
The Sanctuary nightclub is located in the city's west end - an area popular among students A handbag which officers believe belongs to Miss Buckley was found in Dawsholm Park, near Dorchester Avenue, on Tuesday afternoon.
The entrance to the park remained cordoned off on Wednesday morning as dozens of officers continued searching the area. The police helicopter has also been involved in the search.
Miss Buckley moved to Glasgow in February and is a first year occupational therapy student at Glasgow Caledonian University. She was previously a nurse at the Princess Alexandria Hospital in Harlow, Essex.
She arrived at the Sanctuary nightclub with friends at about 23:45 on Saturday and at about 01:00 she told them she was going to the toilet. She failed to return and did not take her jacket.
Her friends have said Miss Buckley had had a few drinks, but was not drunk.
Miss Buckley's bag was found in Dawsholm Park by a member of the public Dozens of officers arrived at the park as darkness fell to continue with the search Speaking on Tuesday, Det Supt Jim Kerr, from Police Scotland's major investigations team, said: "We've traced the man she was with in Dorchester Avenue. He believes he was intimate with her at his flat consensually in the early hours of Sunday."
The police officer added: "From what we can see, she does not appear to be under duress, there's no signs of a struggle or reluctance on her part to leave the club.
"However, that does not mean that something untoward has happened to her at a later stage."
Miss Buckley is described as white and about 5ft to 5ft 2in tall. She has brown eyes and dark hair which had long black curly extensions in it.
When last seen, she was wearing a black jumpsuit with red high-heeled shoes and was carrying a black handbag.
Despite reports to the contrary, Ms Buckley has not been active on social media since her disappearance.
Ms Buckley spend several hours at Dorchester Avenue before leaving at 04:00 on Sunday A heavy police presence has been maintained in the area surrounding the block of flats Police are also keen to find out more about a grey car that was seen on the roads between Milngavie and Drymen north of Glasgow between 11:00 and 15:00 on Monday.
Det Supt Kerr said: "The activity of this grey car on the Monday is something that we're a wee bit concerned about. This car has been seen at various locations on these roads and I want to know why."
He added: "We are gravely concerned that Karen has come to some harm, whether that is down to foul play, criminality or she has taken unwell or had an accident is obviously still to be established."
Her father John Buckley, 62, said the disappearance was "so out of character".
He said: "We are extremely concerned for her. We are desperate to get her back and safe with her family - she is our only daughter, we love her dearly and just want her to come home safe and sound."
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