The NBR still have a lot of insaturation (double bonds), on the main polymer chain.
So still passive of being attacked by the oxygen present on the carbonic acid, or any other oxidant.
Peroxide cured NBR only increases the temperature limit, as the bond C-C have much more energy than C-S or S-S, when you have vulcanization (by the way, Vulcanization must have sulphur... It is a patented, expired of course, but was the process name. Peroxyde is cure...).
NBR can be vulcanized by sulphur or cured by peroxyde,
HNBR will depend of the residual insaturation. Smaller residual insaturation, less double bond on the main chain, and better resistance to oxidation, ozone, etc.
HNBR with certain residual insaturation still possible to vulcanize. Reducing even further, there is no residual insaturation to have the sulphur linking, and you will have to cure wih peroxyde.
It will give you the best resistance to temperature, oxidation and ozone.