In particular, I'm looking for something that can be put on parental PCs that would allow their remote tech support resource (me) to see and interact with their desktop at some time other than when visiting for the holidays or by voice interface over the phone.
Ideas? Recommendations?

LogMeIn! Hands down wonderful. I use this to provide remote support for parents. So easy to use and just works well. Also works across mobile too. No port forwarding!
Google plus hangouts…. Obviously. (Screen share)
+Douglas Boberg An attractive prospect, for a variety of reasons. That would still require "remote hands" to do stuff on the other computer, right?
I've used it for viewing & presenting… But I cannot remember sharing control in either instance. So I'm not sure. You'll have to test it and see. For my purposes it is good enough to replace GoToMeeting.
I do know it has "party" tools that allow you to don virtual clown makeup, mardi gras masks, etc. There's always someone on the call that uses them.
If you've got your parents using Chrome then I'd recommend Chrome Remote Desktop (http://goo.gl/V9bJ9). I use it with my parents. It can be setup either so they would send you a code to allow remote access or you can register the PCs as ones you have control over and login at will (though that requires Chrome to be set to run in the background even when it's not open).
Oh, and +Douglas Boberg, you can do remote assistance using Google Hangouts. Done it that way as well. Works much like Remote Assistance under the old Windows Messenger.
Another vote for LogMeIn. Works perfectly every time.
+Douglas Boberg Oh, I've been using Hangouts for a while, so I'm familiar with the "extras". It would certainly make for an easy way to do the seeing, but if I can get in and actually dig through the machine without having to relate what it is I want clicked on, that might work better.
+Dave Hill, Hangouts will allow a remote assistance as well as desktop sharing. They have to click a button on their end to acknowledge giving you control, but once they do you'll have full mouse and keyboard control.
+Les Jenkins I will definitely have to give that a try, though LogMeIn seems to be pretty popular.