Based on the titles of many live releases, both official and unofficial, it seems that when George & Co tour, they typically go by the name Parliament-Funkadelic. On the Live 4CD set, for example, the majority of the tracks from the 70s are credited to Parliament-Funkadelic. Same with the three unofficial live sets that have come out recently through RoxVox.
But, the two most widely available live releases from the groups that were performed during the 70s - live at Meadowbrook and P-funk Earth Tour Live, are both credited to one half of the usual touring name, Funkadelic and Parliament, respectively.
Was this just for simplicity’s sake, so that the artist name matched the names on the other records? (i.e. at the time there were no studio albums credited to both). Or, at this time were they actually just touring under separate names and only playing songs “belonging” to one band or the other?