I think you are both right. Harris Blitzer and Johnson do not have day to day control of the club. They are happy to cede that to Parish in return Parish gets a wage and a bigger say in voting rights.
Parish has a huge say in the club simply because the others trust him. However if they wanted to it would probably only take 2 of them to push him out of his role as chairman.
It says a lot about him that when Textor tried to throw his weight around regarding transfers he got nowhere. I think it i safe to assume that the others backed Parish over him.