Harpal and Karandeep Singh are desperate for news of their son Manvir, 21, who was last seen by his friend near to the KFC restaurant in Renfield Street at around 4.45pm on Wednesday.
Speaking from the family home in Bishopbriggs, a distraught Mr Singh said: "We just want him to get in touch with us or he could speak to one of his friends, or the police, to let us know he is OK."
Police are carrying out extensive inquiries into the student's whereabouts.
They are studying CCTV images to try and build a picture of his movements.
The family said Manvir, who, celebrated his 21st birthday in February, is popular with lots of friends and is part of local dance group, Punjabi No 1.
He is a third-year student at Glasgow University where he is studying to become a doctor.
His parents fear his disappearance could be connected to the results of his recent exams, which he received around two weeks ago.
Mrs Singh, 41, said: "He never showed any stress to us but we just found out that he failed his exams so I think that he didn't want to come to us because he failed.I think he thinks everyone expects so much of him and he didn't want to disappoint us.
"I asked him about a week ago if his exam results were out and he said: 'no, mum'. But he was just covering everything.
"But we just want to tell him that it doesn't matter, we don't want anything else. We are so proud of him."
On the day of his disappearance Mrs Singh left Manvir in the house at around 8.30am.
She said: "He seemed just normal. He said he was leaving to go to uni at 9am. He never mentioned anything else. We never expected him to not come back."
The former Bishopbriggs Academy pupil, who has a 14-year-old sister Harneet, known as Honey, left his belongings, including his phone and bank card, at home.
Mr Singh, 50, said: "That's what I find very, very surprising. It wasn't planned, his disappearance, because, like me, he wears contact lenses and his solutions are still lying there.
"So he'll need his glasses if he takes his contacts out but his glasses are left here, his Visa is here, his laptop was here.
"The biggest surprise was the phone, because he takes that everywhere.
"It's as if he knew that if he took anything he'd be traced so he's left his possessions in the house."
Manvir is about 5ft 10in, of slim build, with short black hair and a goatee beard.
When he was last seen, he was wearing a navy blue hooded top with yellow markings on it, dark coloured jeans and tan-coloured boots.
He was carrying a black bag.
Chief inspector Rob Hay, Area Commander for East Dunbartonshire, said: "It's not unusual for someone who's been under a bit of stress to get some breathing space but, of course, the longer Manvir is missing the more concerned we get so we would encourage him to contact ourselves or his family.
"We would also ask members of the public to get in touch if they have any information.
"It is a busy street at that time on Wednesday with people leaving work."
l Anyone with any information is asked to contact Police Scotland on 101.