I’ve never seen any sort of consistent rivalry or antagonism between Giants and Jets fans. I’m sure you can find a few of course but generally the fan bases vexing well. They don’t wish ill on each other unlike a plurality of Mets or Yankees fans or a majority of Rangers/Devils/Isles fans.I'm not totally a fan of the Giants but mostly just indifferent. They have won more in my lifetime but I couldn't care less for the Giants as a "city rival". I have a ton of friends who are Giants fans and they're seem of the best people I regularly talk football with.
As a long time Giants fan I will root for the Jets against any team other than the Giants or Saints.
I will admit that even though my distaste for what I see as Brady’s shenanigans is on full display I always have a soft spot for Bellichick who was the Giants DC under Parcells. He was the reason the Giants won those SuperBowls not Parcells in my opinion.