I'll repeast, since a partially incorrect answer has been accepted and lots of people are reporting the same thing:
giffgaff is *not* owned by Telefonica. It is owned by Telefonica UK (a different company).
Telefonica UK trades under the name O2 (or O2 UK) in the UK.
Telefonica UK is owned by Telefonica, as part of the Telefonica Europe division.
Telefonica is a Spanish company with worldwide presence, who operate the O2 brand in a number of countries. They also operate the movistar brand in many other countries.
giffgaff and O2 are *not* siblings, and they are *not* owned by the same company. They're in a parent/child relationship.
The only "difficult" part of this is that O2 is not a company name, but a brand name. Think of it like Sony using Bravia for TVs, or Microsoft using XBox for games consoles, or Apple using MacBook Air for ultra-light laptops.
The other thing to note is that although giffgaff say they are run by their members, giffgaff isn't a co-operative, collective or mutual association. It's just a company with good advertising 😉 (And an ethos)