There used to be three sides, but one's undergoing fission (it would appear).
There are the Houthi's. There's a AQ-related group that's lurking.
There used to be the Presidential Leadership Council. It was the big player in the south, supported by the non-Iran-aligned countries in the area, versus the Houthis (which are Shi'ite and rabidly anti-Israel).
Apparently the new group, Southern Transitional council, is an important member of the PLC that's just decided to play a more important role apart from the coalition it has a seat on. (So the lack of clarity is sort of due to a lack of clarity.) But it's taken ground not only from the Houthis but also from the PLC, so the warfare isn't just against "the enemy" but also internecine.
https://www.economist.com/middle-east-and-africa/2025/12/30/a-lightning-advance-by-separatists-has-reshaped-yemens-civil-war
which is paywalled, for non-pay access there's https://archive.is/gIdEd
Don't like the Economist, there's the Ikhwany https://www.aljazeera.com/news/2025/12/25/saudi-arabia-demands-yemeni-separatists-leave-seized-governorates option. Or the Council on Foreign Relation's https://www.cfr.org/global-conflict-tracker/conflict/war-yemen .
One could see the petroleum issue as just us/US-centered. On the other hand, the PLC needs revenue to continue to function and oil's going to be part of that. Deprive it of funding, the PLC will lose 'umph'. Meanwhile, the STC will get that revenue. (I'm not going to figure out how they see the crude--whether it's through the Sa'udis over land or to them or elsewhere by ship. And maybe I'm wrong and the crude oil's just kept in the ground for now.)