How can guitar pickups hot leads be wired in parallel without causing current problems?
The amount of power (electric energy) generated by these pickups is so small that there are never any "massive currents" present. The currents are simply too small to cause any issue.

When pickups are connected in parallel the generated current in one pickup will divide itself across the other pickups but again the energy transferred that way is quite small. To small even to make other strings vibrate.

You would not want to do the same with for example a couple of generators in a power plant. There the power is such that two generators in parallel can work against each other and cause damage. But for guitar pickups the power involved is simply too small.

To explain how the signals are mixed when the pickups are in parallel I will use this schematic representation:







simulate this circuit &ndash; Schematic created using CircuitLab

I model the pickup element as a Voltage source (the EMF induced by playing the strings), some series inductance (you can ignore this) and some series resistance.

The voltage / current produced by V1 will divide itself across all other elements in the circuit. The same is true for V2. Electrical engineers learn how to do the calculations on this in their first year (I did and I hope that's still true). For this calculation you can determine the resulting voltages of V1 and V2 individually and then later add them up.

What the resulting voltage going into the amplifier will be does not matter much, fact is that the signals are simply added up and end up "everywhere" in the circuit.

Maybe the thing that you missed is that each pickup element has some series resistance, this makes then "play nice" when they are connected in parallel. In the power plant example I mentioned, these resistances are extremely low and that means the voltage sources don not "play nice" as currents can get out of hand.