in the episode called "Poltergeeks" his mom is Barnaby (weird...) and his dad is Mom (even more weird & stupid than barnaby... :P) that's the only thing I got because whenever they do his parents names, it gets covered up :P.
Not knowing the Turner parents' names is just a running joke in the show. Sometimes I think the writters are just too lazy to think up names (I mean that in the nicest, most jokingly way :p) However they must be short considering the names can be said in the half second its takes for a loud truck to pass by. Many beleive the names start with T because of the whole alliteration thing Timmy has with his name. Not a bad theory.
As far as the whole Barnaby thing goes, Somewhere I read that that was NOT meant to be Mrs. Turner's first name, HOWEVER I don't know how credible this sourse was. I myself like to beleive this is her name and that it may have something to do with both her parents being from Ustinkestan (forgive spelling) But I am likely to be wrong.
Fact is, no one really knows and it's very likely we never will. Maybe if we're lucky, and ask the creator enough(ask nicely, and don't bug him too much;) ) he may put it in an episode or make a twitter post or something. Until then, I think it's really just up to the audiences' imagination :)