HSR412 (solid state relay) parallel circuit connection not working?
The parallel configuration can take more current, but look closely and see that it is only applicable to DC.  By "christmas lights", I expect you mean something intended to be plugged into normal house power, which would be AC.

A MOSFET can be a switch when used in forward mode (drain to source positive for N channel).  However, these devices have a diode from source to drain as a byproduct of their construction.  The series case puts the two diodes back to back, so one FET is always blocking when off.  The parallel configuration is only for one power polarity since the diodes will conduct in the other direction.  AC power will be on for at least every other half cycle with the parallel configuration.