Do I need to limit the current on the inputs of a parallel load shift register?
No,
but you should add a pull-down resistor. Otherwise your input is floating when the switch is open. 




  May I know why it's unnecessary? Does the HC165 have an internal resistor or are there other factors I'm missing?


The HC inputs are very high impedance (Which means they have internally a very high resistance). They take hardly any current at all. In effect if you take the TI data sheet it specifies the input current when connected to VCC as a few uA. 

That is also the reason why you should not leave an input unconnected. It needs only very, very little current (read: energy) to switch. It could already switch on the 50/60 electric field from a nearby power rail. 



As to pull up and pull down and switches have a look at this
post which shows you how to use a switch with either.