Reid, 20, from Lenzie, was jailed for six years and eight months, along with Michaella McCollum Connolly from Northern Ireland, by a court in Lima after they were caught trying to bring £1.5 million worth of cocaine into the country in August.
Her parents Billy and Debbie Reid said the punishment was too harsh and would mean Reid spends the best years of her life behind bars. "We think the sentence doesn't sit with the crime that was committed," said Mr Reid. "They're first-time offenders, they didn't instigate it. The other people involved - the bigger players - must be watching all this. They must still not be able to sleep easy at night."
It is believed that under current laws the women are not eligible for a sentence reduction, so they will be forced to serve the full jail term. The family has made a decision not to appeal against the sentence and say their only hope now is that their daughter will be transferred back to the UK to serve the rest of her sentence.
Mr Reid said his daughter made an emotional phone call home after the sentencing, admitting that it had not sunk in yet and telling of her fears about being split up from Connolly in jail.
"This stigmatises Melissa for the rest of her life. She's always going to be known as the drug mule," said her father.
The pair were arrested while trying to board a flight to Spain with 24lb of cocaine in their suitcases. They originally claimed to have been kidnapped at gunpoint by a gang in Ibiza, where they had been living and working during the summer, and forced to travel to Peru but later accepted their responsibility and pleaded guilty.