Do you have a source for this info? Nothing turning up on Google. Search criteria "Commission for Presidential Debates funding".
There was a pbs special a couple of weeks ago that explained the history of the debates and how the CPD was formed after a break with the league of women voters, which used to run the debates. In it, they said the two parties together formed the CPD as a private organization that is run independent of the government so it can make up its own rules. Actually, maybe pbs has synopsis / transcript of the show on their website.