Honestly I can’t see a way for this to be done which isn’t fraught with difficulties.
Firstly your parents would need to arrange an alternative phone service, this would have to be via an Openreach connection, ie BT as VM won’t install another service to the same property - well I suppose they could but the number of hoops which would need to be jumped through, doesn’t bear thinking about! This line would have a new number so they would next need to arrange a number port of the old VM number across to the new line, but I can see issues here as, presumably, the accounts would be in two different names. For this to even have any chance of happening, you must NOT cancel the VM landline or tell them you are moving as that will immediately put a stop to the number port. If (and it really is a massive if), that goes ahead smoothly then by its nature it will cancel your existing VM phone provision but as that is a contractual change it will start you on a new 18 month contract for broadband only. Finally you can move and take the broadband only service with you to the new property.
As you can see there are so many points at which this can all go horribly wrong and, realistically, I just can’t see it working out at all.
So, I can see two possibilities here.
1) Your parents get a new line installed with a new number and spend the time before you move, informing all of their contacts etc of the new number.
2) You don’t transfer the service to the new property, you sign up as a new user which gives you the new user deals. When the existing contracts minimum term ends, then cancel the broadband provision on it - as above this will probably start a new 18 month phone only contract (although I am not 100% sure that such a thing even exists), you will end up losing any VM email addresses you currently have but you will have about three months to make alternative arrangements and let all of your email contacts know of the new address. Technically you will be the account holder for your parents telephone provision and be responsible for paying for it - you would need to work out the practicalities with your parents, also in the event of any issues VM won’t deal with them, on the not unreasonable grounds that they are not the customer, and you would need to handle all of that.
Personally, I’d go for option 1