The Harvard Lampoon, started in 1876 and it's probably the oldest American humor magazine, Little, Brown & Company published a book in 1973 looking back over the years.
Magazine parodies had always been a regular feature, Newsweek, Saturday Review, Esquire, Time (twice) The New York Times had spreads in the book.
It doesn't say whether the 1968 Times was a complete issue or just a front page, the USA Today (below) was an edition with all the supplements. I saved these issues over the years, except for the National Geographic, a copy I missed but I found the cover on the net. If you have a copy you don't want I'd love to add it to my collection.
You can see some general magazine parodies from the last few decades in the January 2014 archive.