How to record my party chat on Xbox?

I do not think there is a straight way to do so.

Solution 1: Get yourself a capture card, that will record every sound out of the XBox. So it will also record party chat.

Solution 2, and cheapest: Back in the 360 days you could have party chat go through your TV speakers. Get a microphone app on your phone (might even be native), press record and set on table. It will record the sound from your TV and so, the party chat also. Only thing, it will also record anything happening else in your room, so if someone is doing a noisy activity next to you, it will also be in the recording.