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

Yes. Because of daylight saving time the gap moves between +5h and +6h.

When is the best time to call Bangkok from Auckland?

Between 14:00 and 17:00 in Auckland, which is 09:00 to 12:00 in Bangkok.

Auckland is 5h ahead of Bangkok. This gap is not fixed: daylight saving time moves it between +5h and +6h during the year.

Auckland

New Zealand

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Bangkok

Thailand

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A 9-to-5 day in each city overlaps for 3h. That is 14:00–17:00 in Auckland, which is 09:00–12:00 in Bangkok.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Bangkok · Thailand	Asia/Bangkok	UTC+7

What is the time difference between Auckland and Bangkok?: Auckland is 5h ahead of Bangkok. This gap is not fixed: daylight saving time moves it between +5h and +6h during the year.
Does the time difference between Auckland and Bangkok change during the year?: Yes. Because of daylight saving time the gap moves between +5h and +6h.
When is the best time to call Bangkok from Auckland?: Between 14:00 and 17:00 in Auckland, which is 09:00 to 12:00 in Bangkok.