In my diocese, the parish has to give a certain percentage of the offertory collection to the diocese. Other collections, like Christmas, Easter, building funds, etc, stay completely in the parish. Special collections go off to that collection. So one could support their parish by giving only to the Christmas and Easter collection but not to the offertory, if they don't want money going to their bishop.

I think the Vatican is funded by the annual Peter's Pence collection, investments, tourism, and wherever else they get their money. Who knows.