The PMI lounge is indeed third-party and run by Aena. It is nothing to write home about, so the OP is missing nothing IMHO.
The inflight "scoff" isn't particularly scoffable either so I would recommend a good go at the No 1 Heathrow lounge on the outbound but the OP will just have to slum it with the (other) tattooed trash in the terminal on the inbound