I've built two CNC machines to run on parallel circuits. Is there a version of Mach 3 for USB (serial) cables?

Yes but you need a USB connected motion controller such as the UC100 from cncdrive.  This has a Mach3 plugin that allows Mach3 to drive it.  I have a lathe and mill both of which use the uc100.  Initially I used them with the same xp pc I used with the parallel port but now use a win10 mini machine.  I didn't even need to redo the ports&pins config!  The uc100 is built into the shell of a db25 connector and plugs directly into the breakout board. The usb cable plugs into a miniature socket on the back of the shell.

Thank you very much. I may come back for more as I investigate UC100.

Happy to help. If you go this route make sure you get the genuine uc100 as there are clones that don't work very well with no support.  There is a good cncdrive support forum and I've found the company very responsive.
http://cncdrive.com/UC100.html

Thank you again. Before you replied I went to their site and saw their strict warning about the clones. But... as far as I can tell, there's no way to purchase one. There's not even a price listed. I sent them an email about it. On another subject but still having to do with my original post, do all breakout boards use a parallel port input so that you either have to have a parallel ported PC or a UC100?

http://shop.cncdrive.com/  ??

It looks like you must be on US east coast?  I'm sure they have agents in the US even if they can't ship direct.

The link you posted above (http://shop.cncdrive.com//) takes me to their products page but the UC100 is not shown there. Just curious-- how did you know I was on the east coast (actually in the SE)?

I found it. It was listed in the controller subcategory.

Hi,
beware that there is many a Chinese knock-off UC100 on Amazon and Ebay. If you are going to buy a UC100 make sure its the genuine
article, the Chinese rip-offs are not a patch on the real thing.

Depending also on your electrical environment you may find USB to be subject to electrical noise and fail in service. A very short USB cable is required, a long cable is inviting
trouble. In this regard Ethernet connected motion controllers are much better, for instance the UC300 which is also made by CNCDrive.

Craig

Still curious to know: Do all breakout boards use a parallel port input so that you either have to have a parallel ported PC or a UC100?