There is at least one parody earlier than the ones we have here today listed for Lala Palooza in Feature Comics #32. (Don't bother with the digital golden age archives. What i've seen of #32 may have the right cover, but the interior seems to be issue #44. There is an odd superhuman in Poison Ivy reminiscent of Flash, but only in appearance.)
Conveniently enough, since we've been doing a lot of Funny Animals lately, the next earliest i've found is Flash Rabbit from All Top Comics #s 1 & 2, written by Pat Parrish (artist unknown) -
Zany #2 brought us Flush Gordon -
Whack #2 remixed the name to Flush Jordan with artwork by William (Alex) Overgard...
...and Humbug #10, Jack Davis had fun mixing it with the paranoid times and gave us Flyashi Gordonovich, Intergalactic Commissar...
...while Wally Wood was hinting towards later work over in Mad #11 with the title Flesh Garden! -
Of course, there have been many more over the years - we're only scratching into the 1950s here. But this post is getting long, so...
So long!
page art by listed creators, otherwise unknown, for All Top Comics #2 1 & 2, Zany #2, Whack #2, Humbug #10, and Mad #11 (1946, 1953, 1954, 1958)