Link video: https://www.youtube.com/watch?v=s4vqgP3P_t4
Hate to burst your bubble but that ain’t a party quest.
You had the hostage quest in your quest log and you teleported to the hostage by right clicking on the warp.
This warped you to the hostage, and because you had the quest to free it, it started the instance.
Another one of your party members then examined the chest to start the red vubbe quest on their side which then started it for you too since you hadn’t done it yet.
These were all just solo quests .
In fact the little party quest seems to just be you seeing the quests that your fellow party members have, as it is enabled by turning quest share on/off on the party settings menu.
Party Quests do not seem to be a thing in this game.
Although there is a tab for party events in the party menu, but that seems to always be empty.
I did a party event once. It was when I had a full party. It was to get free hats and or recipes. We had to go to East Siluiai Woods to kill a boss (which was REALLY hard) but we beat it. Sadly I didn’t SS it.