Here's something that might help people out there like me who are running Mac OSX10.3.9 on a PPC mac.
After downloading the Flash version linked to above, the DMG opens with 2 files: Flash Player.plugin and flashplayer.xpt
There is no installation file at all, and this really had me perplexed.
What you need to do is copy those two files into your Library >> Internet Plug-Ins folder. Restart your browser and Flash should then start working and you'll be able to Picnik again!
Adobe really needs to put these instructions somewhere...

I googled until I finally found something on the Apple Discussions site explaining how to do it.
Hope this helps!
Brenda (Team Picnik)