Here is why: Yemeni government != Yemeni people. The authorities, probably, don't mind letting US drones kill people, simply because it won't be them getting killed.
Imagine the US military firing tear gas and water cannons on protesters in Turkey and saying "It's OK, your government gave us permission". The major problem in the Middle East is that, the governments can get away with not giving a fuck about the people.
I think you're having reading comprehension issues. It's not just that the gov't are requesting help. The reason we are involved in Yemen is because the Yemini gov't is incapable of exerting it's own will b/c they are weak and poor. Their interests are also strongly tied to our interests - so that's why we intervene.
Your example has is not at all analogous. The Turkish gov't is not incapable of tear gassing and water cannoning protesters, so they fail the first criteria of being incapable of carrying out their will. The example also fails the second criteria for intervention - that it's strongly in the US's interest.
"Yemeni government != Yemeni people"
Who else can you appeal to in Yemen to get permission to operate drones?
And are you seriously suggesting that the Yemeni people are supporting Al Qaeda? If they are, then they become the "enemy".
The way it works for other nations is we tell them: 1 - You will fight Al Qaeda in our country 2 - If you are incapable, we will help you do it by providing military aid. 3 - If you are still incapable we will us our own armed forces to make sure you are able to. 4 - If you are not willing to fight Al Qaeda or are actively defending them they we will intervene anyways.
The last point either leads to the case of the Taliban where we destroy them because they are defending Al Qaeda, or to Pakistan where they are forced to accept our help.