Below is a picture of what John described. The DC line is fused and I don't think there could be enough current getting to the park cable to get it warm much less melt.
Unless your cable shield is somehow shorted to 120 vac the problem just about has to be with the cable coming from the park. There is supposed to be a small capacitor in the feed from the park amplifier/distribution box to the pedestal and there should be no measurable dc or ac voltage on it. The same is true of your cable.
The only instance where a dc voltage should appear on your cable is if you have a satellite receiver connected. That voltage would be 13 to 18 volts and is a high impedance source with current limited to a very small amount and could not cause any heating if shorted. Since you don't have a satellite receiver connected that is a moot point anyway.
