Is the basis of parity functions the only orthonormal basis for Boolean functions? Is there another orthonormal basis of functions for Boolean functions? Or, more specifically, besides the parity functions, is there another explicit function (which is common and has a name) that can form an orthonormal basis for Boolean functions? 

If not, why not?
I don't think parities are the only orthonormal Boolean basis, for example Paley basis provides a Boolean orthonormal basis in some cases. It's natural to ask if such bases, interpreted as Boolean functions, have any interesting applications.