Hi John,
I haven't been able to reproduce your problem with IE9. I did notice that the error message you attached said "This file is too large to import and display in your web browser (Internet Explorer 7 or 8)." Just want to make doubly sure you're using IE9. I'll try and find out from ArcGIS Online people here if they're aware of problems with IE9 failing to import shapefiles. In the meantime, see how things go with Chrome or Firefox. If you like, send us your parks shapefile (and the others too if you want) as an email attachment to UnderstandingGIS at esri dot com. Then we can test your exact data to see if we hit the problem, too.
Thanks,
Tim Ormsby
Esri