I've had a semi-rough idle for a couple years now but it's started to stutter at idle and acceleration.

Scanner is telling me P0306, #6 cylinder mis-fire and P0420 Catalyst system below threshold.

Are the two codes related?

New parts:
Fuel pump (ACDelco)
Fuel filters
Plugs
Wires
Radiator
Heater Core
Starter

 

Yes, they are related, 306 means what it says, #6 is not firing, the fact that you are getting 0420 probably means its a spark issue and that cylinder is not burning its mixture and sending raw fuel into the exhaust, 0420 basically means that the final O2 sensor after the catalytic converter is sending information of a rich condition or the converter is not performing properly.

Check the distributor cap terminal, and look at the plug wire terminal for #6, they should be shiny silver, if theyre black you need to replace the cap and bad wire. These distributor caps can crossfire internally when they get old, if you replace and ignition component on this engine you MUST use factory brand name parts (AC Delco or Delphi) everything else is junk, if you just put a cheapo brand set of wires on it a month or two ago and now have this problem then you should look there first, all parts are NOT created equal!

Also if the p0420 has been there for some time then chances are pretty likely that the catalytic coverter has been fuel contaminated and damaged by this time, best thing to do is not drive it all til you get this misfire taken care of.
This could also be caused by a dead or plugged injector that made a lean condition causing the converter to overheat and meltdown internally.
Many variables, some idle and 2500 rpm fuel trim numbers would help big time in determining what direction to take
For now look into the ignition system and let me know what you find
Good luck!