Looking at the posters/handbills they were mostly credited Parliament-Funkadelic (until 1981). But up to around 1975 they were sometimes credited differently.
I don’t believe that was the case.
What I always thought was that the P.Funk Earth Tour album was credited to Parliament because that name was the most famous at the time.
And that Live at Meadowbrook got credited to Funkadelic just felt right, with it’s guitar heavy sound and all. But I don’t know what name that show was actually credited to. There was another show nine days earlier (on the 3rd of September 1971) that was credited to Parliament Funkadelic…
But writing this, it hit me that it could also be a record company thing. Casablanca released the Parliament stuff and Westbound/Ace released Funkadelic. Maybe they couldn’t or didn’t want to use the other name.