There is NO WAY Chef, or anyone known by the 4 children, is Cartman's dad.
Wanna know why???
One very simple quote:
"Why don't you stop... dressing me up like a mailman, and making me dance for you... while you go smoke crack, and HAVE SEX WITH SOME GUY ON MY DAD'S BED THAT I DON'T EVEN KNOW."
See? Again: "THAT I DON'T EVEN KNOW"
Now, the fact that Mrs. Cartman dresses Cartman up like the mailman can mean only one thing.... THE MAILMAN IS ERIC'S DADDY!!!
Eric wouldn't know him because he is obviously in school when the mail comes, except for Starvin Marvin; but even there he didn't see him because they drove off when the kids opened the door!
Now, I ask:
Who's with me??