" Siding with Yemen " is a bit rich. The writing is on the wall for the current regional war to expand into a proper world war. They don't want their infrastructure, be it shipping or oil related to be hit by missles.
It's one thing to carry out an air campaign halfway across the earth where your enemy cannot reach your homeland. It's a bit different when the war is in your backyard and not on the television.
Same reason why Saudi Arabia is so squeamish about this whole air raid campaign. Both nations host American bases and troops. They are the logical first targets for the Houthi's to attack. Signing off on this would be stepping foot into the ring as a meatshield, with little to gain from this charade.
Yemen has already been bombed back into the stone ages, infrastructure wise. More airstrikes will not break them.