As said above, your fathers neighbour needs your fathers permission to build on to the party wall. If he doesnt get it, he can just build an extension inside his own boundary, subject to planning, as long as no part of it encroaches on to your fathers property.
As to whether your father should give permission or not, I'd agree with the answers above in that he would probably need an engineer to advise him on whether the wall itself is strong enough for the development proposed, whether the proposed extension would affect him in any way, whether it be by blocking light, how rainwater is to be disposed of etc. An engineer has a cost though.
If your father doesnt mind the actual proposed extension, could he ask the neighbour a) to pay for the cost of him consulting an engineer to check it out for him and have the neighbour agree to modify his proposed extension as per any recommendations of the engineer and b) get agreement in writing that in future should your father want to build such an extension that he has permission to do so?