The 15-year-old has not been seen since she failed to turn up on Friday at Bishop Bell C of E School in Eastbourne, Sussex, where 30-year-old Jeremy Forrest, an amateur singer, teaches.
Megan's mother Danielle Wilson and stepfather Martin Stammers spoke at a Sussex police press conference that was translated into French as detectives there search for Mr Forrest's black Ford Fiesta.
Holding back the tears, Ms Wilson said: 'Sweetheart, I don't care what you've done or why, I just want you home.'
Mr Stammers added: 'I just want you to come home safe and well. We are a family unit working together. Strength is coming from within.'
Both said that Megan's disappearance was 'so out of character'.
In a direct appeal to Mr Forrest, Sussex police Chief Inspector Jason Tingley said: 'Do the right thing and make contact with us.'
He added that return ferry tickets from Calais to Dover on Sunday had not been used.
Terry Boatwright, executive headteacher at Megan's school, expressed deep concern and shock at events.
'The thoughts and prayers of us all are with Megan and her family at this time,' he said.
'We would urge her to get in touch with her family to let them know she is OK and we would echo the police's call for anyone with information to contact them. Clearly, everyone's main concern is Megan's safe return home.'
Mr Boatwright added: 'As a matter of urgency, the school and local authority are investigating exactly what has happened before deciding what action needs to be taken.
'We are in close contact with police who are at a critical stage of their investigation and, for that reason, we cannot comment further at this stage.'
Jeremy Forrest, Megan Stammers Hunt: Jeremy Forrest pictured on his wedding day
Police have released details of Mr Forrest's car, registration GJ09 RJO, in a bid to find the pair.
In May, under his stage name Jeremy Ayre, Mr Forrest wrote on his personal blog referring to a 'moral dilemma'.
'About a week ago I had a bit of a moral dilemma to deal with, both internally and externally. And the overriding question it left me with was this: How do we, and how should we, define what is right or wrong, acceptable or unacceptable?' he wrote.
'I came to a few different conclusions, mainly that actually we get a lot of things wrong, but at the end of the day I was satisfied that if you can look yourself in the mirror and know that, under all the front, that you are a good person, that should have faith in your own judgment.'
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