My impression, after having looked around a bit on the Internet, is that a vast majority of citings give it to Parr.
Here's a theory. I think it could be more than just a coincidence that it is attributed to Parker, because on several lists that were alphabetical by the author's last name, viola! Parker was just before Parr.
I have noticed that it is easy to mistakenly take the name just before a quote as the attribution (as some lists do it that way, but most list the name after the quote).
I do a lot of "tracking" down quotes (sort of a hobby I guess) and I am often amazed that a good one-liner will be attributed to someone that has absolutely no other claim to fame except that one-liner. How do these things come into existence is a mystery I know I will never solve.
Here's one webpage that looks fairly reputable that gives it to Parr.
For a chuckle, check out these "folding" quotes