The 24-year-old's "desperate" parents made the trip from Co Cork to Glasgow where they made an emotional appeal for information on their vanished daughter.
A major police search including helicopter support last night moved into parkland in the city, near where her handbag was found dumped in a bin.
Attention also moved outside Glasgow where a grey car has been spotted "a number of times since Monday" and may be linked to the case.
Officers admitted they were "extremely concerned" for the young woman who has been described as a "sensible girl".
The investigation took on a new emphasis following the discovery of Karen's bag, near the house where she spent several hours with a man in the early hours of Sunday.
But police stressed that the man, who was seen with her on CCTV outside a Glasgow nightclub, is not a suspect in the case. The pair were spotted as Karen left The Sanctuary on Dumbarton Road in the city's West End shortly after 1am.
Detective Superintendent Jim Kerr has said that Karen "wasn't under duress" when leaving the popular nightclub.
He added: "The CCTV shows Karen talking with a number of males in the nightclub.
"There were two that we wanted to talk to, and we tracked them down and they are helping us with our queries."
One of the men told the police that Karen came home with him, but she left his flat at around 4am and was planning to walk home to Hill Street in the Garnethill area.
Karen's mother, Marian (61) and father John (62) made an emotional appeal at yesterday's press briefing in the Scottish city.
"We just want Karen home safely," said her mother. "We are desperate, she is our only daughter.
"We love her dearly and if anybody has any information, please come forward. We would dearly appreciate it."
Her father added: "Like Marian said, if anybody is holding Karen or anybody knows anything about her, just please return her.
"Just give her back, we love her to bits. We're devastated. Thank you."
The couple travelled from the north Cork village of Mourneabbey while her brother Kieran (28) is on his way from Sydney to be with his parents. She has two other older brothers, Damian (32) and Brendan (26).
Det Supt Kerr added that the CCTV footage didn't reveal any violence or struggle between Karen and the man she left with. It appears the man left the nightclub first and was followed outside by Karen - who did not bring her coat with her.
He drove Karen to his apartment on Dorchester Avenue.
He claims it took him 10 minutes to make the two-mile journey back to his home.
Police said that, according to the man, they were intimate before she left at 4am.
Det Supt Kerr said that it "would not be a rare occurrence" for her to walk home after a night out.
Despite not being named a suspect, police cordoned off the property.
Later, the occupational therapy student's bag was discovered close by.
It is believed a member of the public spotted the item in a bin near an entrance to Dawsholm Park. A secluded entranceway was sealed off by police a short time later.
Air support units and foot patrols scoured the park yesterday, but their search had uncovered nothing as night fell.
Det Supt Kerr said that Police Scotland are still viewing the investigation as a "missing person" case and are following a number of leads.
"We have received a large number of calls reporting sightings of Karen and also a large number in relation to sightings of a suspicious car.
"We are not yet sure if the two are connected but want to rule them out," he added.
Police say they are interested in "following up on" a grey car spotted in remote locations outside Glasgow city.
"It has been seen in a rural area around Milngavie a number of times since Monday; we are not sure what it is doing out there."
Karen's friends raised the alarm at 5pm on Sunday after they had failed to hear from her.
They had been drinking at home before heading to the nightclub just before midnight on Saturday, but stressed that Karen was not drunk when she told them she was going to the toilet before she disappeared.