They didn't start off backed by Iran or Russia. 

They were forced into Iran's arms because Saudi Arabia attacked them. The US backs SA and it looks like Israel has some kind of control over SA too. The houthis needed an ally. Iran is the only ally available. This situation is silly - if we hadn't been bombing them, they might've already finished their civil war, and then we could have opened diplomacy with them. Similarly, if we (US + SA) hadn't bombed them, they wouldn't have needed Iran. For us, houthis winning didn't need to be any kind of change from status quo. For SA it also didn't need to be a change, but since they decided to go bomb crazy, America wanted to participate. 

The media always leaves out this super important detail - we attacked them first. I'm not saying they're good guys - I don't think they are - but if someone attacks you, you have every right to fight back.

Russia is only a factor for the warning power. There's no chance they're contributing anything material. 

 Good summary