This is the problem with Iran's strategy, and why it is completely alienated at the world stage.
Iran does not want to play with the international rules based world order developed by the US/West, which is why it is sanctioned and not part of the international banking system.
Iran is blessed with natural resources, but has to adopt different ways to get over this impediment. Hence, it is highly dependent on China; which is part of the international rules based world order, and who's economy is intrinsically tied to the US.
Iran cannot afford to lose China's partnership, but China can lose Iran as a trading partner.
Now coming to Pakistan, Pakistan has historically very close ties to Saudi Arabia. Iran is a neighbor, and Pakistan will make sure it does not antagonize Iran in the current war. Pakistan has been very transparent that the security of the KSA is very important to it, while also supporting peace and normalization of the Iranian economy/state.
About the Houthis, yes, they have a right to do whatever they want against the KSA-backed PLF, or KSA. Pakistan has said it is simply providing defensive support to the KSA, nothing more, nothing less. It is not directly involved in operations inside Yemen.
Iran activated the Houthi card to take over the Bab-al-Mandab, to relieve pressure from the SoH, as well as to pressurize the KSA and the European nations that get their supplies from there.
I do think Iran will be going too far if it involves itself directly in the KSA by launching missile strikes into the KSA. Because as far as I know, the US bases in the KSA are not being used to strike Iran right now. Iran will provide support to the Houthis, but it is a non-state actor which eventually has a limited force (in comparison to various countries).
In short, I think Iran has opened too many fronts, and it is very isolated on the world stage; because it likes to believe it sees things as "black and white" (full sovereignty), but thinks Pakistan does not. That's a very incorrect way to look at things; the reality is, Iran does not know how to manage world relations; whereas Pakistan does.