Dear Paul,

I realize the problem is quite deep, here.
Anyway, as they may be useful, I attach the two functional connectivity matrices obtained using the two methods (one subject, two concatenated 7.5 minutes resting-state runs).
You may want to know that the two approaches produces similar - but not identical - outputs (the 2D correlation coefficient between the two matrices is 0.83).
There is a medium-low correlation (r=0.32) between the parcels' size (in voxels) and the change of parcels' mean connectivity between the two methods. Parcels' size is also correlated to the shift in the degree centrality (r=0.47) and betweenness centrality (0.31).
Thus parcel's size may have an impact. Probably, "driving the averaged correlation toward zero" may be stronger if more operations are computed (i.e., more voxels).

Just to be clear. With "trust a parcellation" I didn't mean something like "trust by faith". Parcellation~=Revelation.
There are some parcellations which are more similar to an undersampling, identifying circular-shaped, equally-sized parcels. In this case, I think that the two approaches may be more or less equivalent. Other parcellations may have been aimed at identifying areas which encompass voxels which are likely to have common functional signatures. In this case in my opinion the approach average->correlation makes more sense.