1. Download this booklet - it contains a lot of information about the Party Wall Act
http://www.communities.gov.uk/pub/411/PartyWal letcAct1996explanatorybookletPDF597Kb_id113141 1.pdf
2. Unfortunately, so far as I can tell from the booklet, there seem to be no enforcement provisions to deal with the sort of thing that has happened to your parents.
3. The surveyor clearly has a conflict of interest and should never have agreed to act as the surveyor for the purposes of the Act. He is presumably a member of a professional body such as the Chartered Institute of Surveyors & these bodies normally have disciplinary procedures. Also the estate agents firm he works for may be a member of an estate agents body that has such procedures. You need to try to find out what the relevant bodies are. Then put in writing to the surveyor the precise complaints and ask for confirmation that the agreed award as set out in his documents will be implemented, including the whole of the fence. If the surveyor has a manager (such as the overall manager of the office) in the agents firm send this letter to him instead of the surveyor, wording it as a complaint about the suveyor. If this does not result in the necessary action, make complaints to the professional bodies.
4. As regards the fence, it might be that the planning application plans show the fence going the full length. Check the plans at the local council office & talk to the planning officer about the whole problem. If what is happening is contrary to the plaaning consent he should be able to help.