Self moving back from France advice needed

There is no duty to pay on the movement of personal goods across borders within the EEC so from that point of view you should have no problems at all.  See this link:
http://www.hmrc.gov.uk/customs/arriving/moving.htm

If you are stopped by Customs some of them can get a bit "jobsworth" so a letter might not be a bad idea, although strictly-speakng it should not be necessary at all, and in the end of the day, anyone can write a letter, can't they?  If you are worried about anyone thinking you might have stolen the furniture etc. I would not worry.  Customs aren't the slightest bit interested in that kind of thing because stolen goods carry no duty - she said, cynically!  :-)

I am assuming your parents don't have any restricted goods, such as a shotgun.  If there is a shotgun you would need to phone your local UK police station and ask to speak to the Firearms Officer. He or she will tell you what needs to be done, but unless it is a very valuable one bringing it back might prove more complicated than the gun is worth.

Even if they have a large collection of wine this is unlikely to exceed what is considered "personal consumption" amounts but if you had to do this kind of trip for more than one relative this year (!) it might prompt questions.