Nearly 50% of people believe that half or more of MPs are corrupt according to a BBC/MORI poll (June 2009, in light of The Telegraph's expose of MPs expense claims). You can see the BBC report here. "It also points to strong support among the British public for taking away Parliament's power to police its self."
The appointment of Parliamentary Supervisors seems to me to be a step away from democracy, unless, of course, they are elected, but there is nothing to suggest that that is what is on the public's mind. And even if the supervisors were elected, who would supervise their pay and expenses? Would they cook up a cosy deal with the MPs for mutual benefit? And, by the way, what about the expenses of the Lords in the higher house?
My preference is for public information on expenses and Interactive Democracy, so that we can keep politicians in check.
In the ID system the public could call for a General Election if Parliament behaved scandalously or we could just dismiss corrupt MPs one at a time. Or perhaps make other suggestions to bring them to justice: a police investigation for example.