Military strikes on non-state actors to discourage further attacks with no significant presence on the ground? Probably not an actual war.
Full-on invasion of a sovereign state using a massive proportion of your total ground forces to penetrate their borders and massacre their citizens? Probably a war.
This doesn't seem to be a clear description. The Houthis look like they control the largest territory in terms of population and cities, but the internationally recognized government factions are in control of more land. And are also recognized as the ruling authority.
Certainly, they aren't a small and trivial group, but they're also not a sovereign government. It's a large revolutionary group. A good comparison is probably the US revolutionary war actually -- I don't think I'd say the US colonists were the biggest political faction over the British. They were certainly powerful and ultimately won, but they weren't the controlling authority of the region.
Nobody cares about some desert land. If you have the population you rule the country.
What is internationally recognized is not relevant. If Iran recognizes something we don't care either.
The fact is that America is not just conducting some "special military operation" in the middle east. They are actively at war defending israel's Genocide.
I'm not disagreeing that the US is doing that. I just don't know the veracity of calling the Houthis the biggest political faction.
I also don't know that war is the correct term, but that's more of a semantics difficulty of the 21st century. Supplying arms vs conducting airstrikes vs ground invasions -- I agree with what you're saying overall though. The US is helping Israel, and Israel is committing genocide.
I really don't think this is germaine though to the topic of Russian propaganda in the Ukraine war.