I honestly don’t think it’s possible to remain friends after one person in the relationship feels more than friendly. And added to that, you have to not only let her down but also her parents which creates an awkward situation.
I think you need to give her time and wait until she gets over you before carrying on with your relationship - if this friendship is important to you. Otherwise - cut your ties. Another option to help her mend her broken heart – introduce her to someone – encourage her to fall out of love with you and in love with someone else. If it works out - she'll be grateful to you and name her firstborn after you - if it doesn't she'll hate you for ruining her life ;)