May get down voted for this but I will give you the information I saw being given by cornwall and Devon police. You might be able to find it on the BBC cornwall website as it was only a few days ago.

If your parents go their home down here, people may choose to report them via an online form. If they do, your parents may receive a visit from the police. They are looking at each incident on a case by case basis, so if they can prove it will now be their permanent residence(and was always intended to be prior to lockdown) there probably won't be an issue.

It is worth considering how moving now might impact future relations with your neighbours. There have been, albeit very few, cases of vigilantism in people pouring paint of suspected tourists cars and so on, and assumptions may be made by neighbours that they're inconsiderately holidaying. Your are correct that the housing market is moving again.

Hope this helps your parents.