No and HELL NO.

One simple correction is to use Type SER cable to completely replace the existing sub-panel feed.

OK- Thanks for these replies.
One last question.
If I remove all 120V breakers and just devote this subpanel to 240 -2P loads (no neutral and thus no neutral current) does it still need a neutral wire from the main to meet code?

Here is a pic- obviously not done by a good electrician.

WHOH Not done by a good electrician? Not done by an electrician at all. In addition to the improper grounding, it's only a matter of time until that piece of NM running through the naked hole wears through and electrifies that panel (hopefully tripping a breaker in the process but given the dumbass grounding procedure, there's no indication that it will happen before killing somebody).

I fear that if I had more of that picture, I'd find more dangerous, stupid, and otherwise illegal stuff going on.

Get thee to a qualified person!

Yes ronnatalie, That's what I meant-
There is one other 20A cable coming in the same way lower down in the box.
Can anyone give an opinion about my last question?

If there are no 120V (or other loads requiring a neutral) you MIGHT get away with it, but frankly I suspect an inspector will give you a hard time on a panel wired as such because nothing stops some unsuspecting person from adding such a load later not realizing the goofball cob job this is.

We removed the subpanel and used the SE cable with a
60 A disconnect at the A/C location to serve the AC with 240V.
Changed breaker from 100A to 60 A in the main panel.
Other circuits were rewired to the main panel.
Looks better now.

Thanks to all who replied.