2012 MacBook Pro running Catalina (10.15.7) trying to connect to 2010 iMac display running Sierra (10.12.6). Help please!

HDMI won't work...

1. Connect the two computers using the appropriate Mini DisplayPort or Thunderbolt cable. 
2. You can use more than one iMac as a display, if each iMac is using a Thunderbolt cable to connect directly to a Thunderbolt port on the other Mac (not the other iMac). 

Use your iMac as a display with target display mode - Apple Support

I have successfully enabled Target Display Mode on those two exact models with those macOS versions, and all you need is the miniDisplay cable. How to invoke Target Display Mode is in the article DBAqua linked. You use keyboard commands, not a display system preference.