As Jetstar Asia is technically a Singapore airline with Singapore AOC and access to Singapore's air traffic rights, QF can use this vehicle to fly all over Europe (Singapore has unlimited 3rd and 4th freedoms with all European countries except France and Germany I think - but since Singapore has signed open skies agreements with nearly all EU countries, SIN-EU open skies isn't too far off the agenda)
Because JQ has no rights to fly between Singapore and NZ, 3K is the designated Singaporean carrier to fly this route, as SIN and NZ have full open skies.
I suspect SQ will stop FCO and ATH once 3K starts those services. They've languished at 3 weekly for the past 25 years.
SQ is now being kept on their toes because they must share rights with JQ. Eg they had to reinstate the morning flights to Chennai and Bangalore using Silk Air, if not they would have lost those slots and they could have been reassigned to 3K.