Just because there is a loose alliance and significant military support between several organizations doesn't mean that they all do the main party's bidding, much less that they only do the main party's bidding. (To whit, note that the US can't really order NATO to do much of anything).
I suspect that Iran isn't in any sense ordering the Houthis to do anything, and the idea to attack shipping is entirely germinated from the Houthis themselves (largely as a form of "showing support" for Palestinians, even if, as a few people have noted, this action only hurts them while providing them no meaningful aide). Largely, this is because it's colossally stupid for Iran to push for a provocation that is meaningfully likely to see all the major world powers (except for maybe Russia; note that China is going to be on the US's side here) create a coalition to eliminate a troublesome menace.
This probably is the case, but I think it's worth saying that it doesn't actually change Iran's exposure here, because I presume the rest of the world believes that the IRGC can stop this (or at least drastically scale it back) any time they want. If you were the UK Secretary of State for Defence, where would you rather engage? Houthi Yemen, or anywhere that Iran has an interest?
I suspect that Iran isn't in any sense ordering the Houthis to do anything, and the idea to attack shipping is entirely germinated from the Houthis themselves (largely as a form of "showing support" for Palestinians, even if, as a few people have noted, this action only hurts them while providing them no meaningful aide). Largely, this is because it's colossally stupid for Iran to push for a provocation that is meaningfully likely to see all the major world powers (except for maybe Russia; note that China is going to be on the US's side here) create a coalition to eliminate a troublesome menace.