Similarly, sectors on EI will not earn any Avios in EI's own program (AerClub) unless the ticket is issued on EI stock...so I wouldn't expect those flights on this ticket to earn anything in any programme....unless the website you are using is actually selling you separate tickets cobbled together, which is actually a possibility.

I'm 99% certain NZ doesn't codeshare on ANY Aer Lingus flights, which also makes me suspect that this is an agent who is packaging different tickets together, rather than selling a "true" interline ticket....