[hackerlight]

The Iranian regime is separate to Iran the people/country. The interests of the two can diverge.

Supporting the Houthis achieves a few things. It ties up Saudi Arabia, one of Iran's regional rivals. It gives Iran a level of force projection over the crucial gulf shipping route, which is leverage. It gives Iran the ability to have the US or Israel struck without giving the US or Israel the easy narrative ability to strike Iran directly in retaliation.

Also, destabilizing US-aligned Arab countries is in Iran's benefit because secular US-aligned dictators are quite anti-Iran and compliant with the US. For example if the Houthi activity leads to economic troubles in Egypt, it could lead to a popular revolution. The US friendly regimes gets ousted by populists (most likely Islamists win at the end), who will be more pro-Iran bloc than compliant with the US.

Another thing to note is that Iran, like Russia and China, is a revisionist power. They are not a status quo state. The TLDR of this is that they hold a grudge and a lot of their energy is dedicated to changing the world/regional order.