Does the time difference between Auckland and Paris change during the year?

Yes. Because of daylight saving time the gap moves between +10h and +12h.

When is the best time to call Paris from Auckland?

There is no overlap between standard working days, so one side has to call early or late.

Auckland is 10h ahead of Paris. This gap is not fixed: daylight saving time moves it between +10h and +12h during the year.

Auckland

New Zealand

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Paris

France

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A 9-to-5 working day in Auckland does not overlap one in Paris at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Paris · France	Europe/Paris	UTC+2

What is the time difference between Auckland and Paris?: Auckland is 10h ahead of Paris. This gap is not fixed: daylight saving time moves it between +10h and +12h during the year.
Does the time difference between Auckland and Paris change during the year?: Yes. Because of daylight saving time the gap moves between +10h and +12h.
When is the best time to call Paris from Auckland?: There is no overlap between standard working days, so one side has to call early or late.