I think it's time to start unwrapping the wiring. Can you feel it getting hot anywhere before the fuse blows. Perhaps some wires have melted together in the harness.

I have had similar problems where previous owners have done very dodgy wiring.

PVC wrap could be hiding all manner of horrors.

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68 Morris Cooper S Mk1 (*ex 78 1275 LS 4th last built, 70 Morris 1500 OHC & 70 MiniMatic)

I'm only having a guess here but it sounds most likely that the insulation on the 12V suppling power to the parker circuit has shorted out on the body somewhere. And when you tap the light the 12V earths out creating a short and blowing the fuse shortly after working for a few seconds. I had a look at my diagrams and think that the wire you are chasing would be the RED one. I'd have a look for insulation break down on the wire as it passes through the different bulkheads. The earth should be fine. On the parker circuit the globe earths directly to the body through the mount. The 3 wires Phillb mentioned are the 12V GREEN WHITE for the Indicator, BLACK EARTH and 12V RED for the Parker according to my diagrams.

I'd start by removing the light assy from the panel - checking the insulation on the RED wire to see if its damaged or find evidence of the short. Like Phillb said, start at the globe and go inwards.

Can't back any of this up with facts, it's just what I would do first if I was doing it. I'm guessing the fuel gauge stops working because it uses the same fuse not because it blew the fuse. Same as the horn I think. Try to see if you can toot with the blown fuse. It does take bugger all of a connection to make a short. Even a loose single strand of copper can cause a short.