The Houthi takeover prompted a Saudi-led coalition to intervene months later and the conflict turned into a regional proxy war between Saudi Arabia and Iran, with the US long involved on the periphery, providing intelligence assistance to the kingdom.

Source: https://jerseyeveningpost.com/morenews/worldnews/2023/09/20/saudi-arabia-praises-positive-results-after-yemens-houthi-rebels-peace-talks/

While it is true that Iran and the Houthis are friendly with each other, that does not mean the Houthi takeover of the government was backed by Iran.

Source: https://original.antiwar.com/trenton_hale/2023/02/08/how-the-media-ignores-yemen/