Doctors treating the 12-year-old are likely to face pressure to report the pair to police and social workers after criticism from nutritionists concerned that the couple's commitment to veganism took precedent over their daughter's wellbeing.
The girl, who has eaten no meat or dairy since her birth, is said to be suffering from a severe form of rickets, as well as several fractured bones.
Rickets is usually caused by a lack of Vitamin D and causes a softening of the bones in children, which can lead to deformity and fractures. It is common in developing countries where malnutrition is widespread.
The consultant paediatrician treating the girl at Glasgow's Royal Hospital for Sick Children, Dr Faisal Ahmed, refused to discuss the case.
But he said that the risks involved in forcing a child to adhere to a vegan diet ought to be made clear.
A leading nutritionist, who did not want to be named, said: "In most instances, the parents who are imposing this very restrictive and potentially hazardous diet are not themselves brought up vegans."
"They are imposing on their children something...which we do not know enough about to know it is safe."
Jonathan Sher, head of policy at Children in Scotland, which represents 400 child welfare organisations, said social workers should intervene in cases where a child's health was being put at risk by their diet.
Glasgow city council said the incident had not been referred to its social work department.
In 2001, a vegan couple from west London were sentenced to three years' community rehabilitation after admitting they had starved their baby to death.