I want to use my 21.5 inch, Mid 2011 iMac (High Sierra 10.13.6) as an extended display for my 24 inch, M1, 2021 iMac. (Ventura 13.4) I purchased a Thunderbolt 2 cable and a Thunderbolt 3 to Thunderbolt 2 adapter to connect the computers. No luck!

That setup is not supported by Apple. The iMac cannot be used as a monitor for any of the newest Macs and Target Display mode is no longer a feature of macOS, having been deprecated several years ago.

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

Per the link above:

• Your target iMac (monitor) must be using macOS High Sierra 10.13.6 or earlier. 

• In all cases the source Mac that you're connecting FROM must have been introduced in 2019 or earlier and have macOS 10.15 Catalina or earlier installed.

That 2011 21.5" iMac only has a resolution of 1920x1080 pixels. Because it has Thunderbolt 1, you can only use TDM over Thunderbolt 1. If Apple did not arbitrarily restrict the source Mac, then you COULD use it as a monitor for a modern Mac with a TB3 or TB4 port, but you'd need a $30+ Thunderbolt 1/2 cable, plus a $50 TB3-to-TB2 adapter.

Given that new 24" 1920x1080 pixel monitors – even decent ones – are now pretty cheap, spending $80+ to use a 12-year-old iMac as a substitute for a new display isn't particularly economically attractive. The restrictions on the source Mac are just an additional row of nails in the 2011 21.5" iMac's TDM mode's coffin.

Where the new "2019 or earlier" and "Catalina or earlier" restrictions really hurt are with the 27" 2009 and 2010 iMacs – the ones with 2560x1440 resolution that could take Mini DisplayPort input.

hcsitas wrote:

Not possible, unfortunately. Use a regular monitor instead.

Not possible, unfortunately. Use a regular monitor instead.