Using Late 2013 iMac as monitor for Mid 2014 MacBook Pro

With the specific configuration you are asking about, there are at least two reasons why it wouldn't work.

To use your Late 2013 iMac as a Target Display for your Mid 2014 MacBook Pro, you would need to

• Downgrade your Late 2013 iMac to High Sierra or earlier
• Downgrade your Mid 2014 MacBook Pro to Catalina or earlier
• Connect the two using a Thunderbolt 1/2 cable (not a Mini DisplayPort cable)

See: Use your iMac as a display with target display mode – Apple Support

You could perhaps downgrade by preparing and running from alternate, external startup drives, but you would need to downgrade both Macs, if you wanted things to work at all.

In the long run, you would be better off just purchasing a proper external display as a replacement for that 2013 iMac. In your current configurations, you cannot do Target Display mode between the two, even with a TB 1/2 cable.