The deployment of nuclear missiles to Cuba _was _a response to nuclear missiles being deployed to Italy and Turkey. This was also the prevailing wisdom during the cold war.

But thanks to information which was only released after the cold war, we now know the Soviets had already stationed nuclear weapons in East Germany, and that the US knew that they had deployed weapons to East Germany. In 1956 Kruschev had also made the infamous (but arguably misinterpreted) We Will Bury You speech. This was seen in the West as an explicit nuclear threat. The article above:

"He threatened in 1956, at the time of the Suez Crisis, to use nuclear weapons, and he very much regarded his threat as successful, so he saw nuclear blackmail as a valuable policy initiative and he needed, therefore, the nuclear missiles to back that up," he says.

The article mentions that due to increased tensions (the aftermath of Suez is less talked about because it was eclipsed by Cuba), Kruschev withdrew these nukes in 1958, but crucially the CIA did not know that till 1961.

Ie. Cuba was a response to American missiles in Turkey/Italy. But Turkey/Italy was a response to previous Soviet threats and deployments. The Soviets deployed missiles in East Germany because Kruschev (arguably correctly) had come to the conclusion his threats at the time of Suez had worked.

A reminder on Suez/1956:

During the summit in Paris, Mollet commented to Adenauer that a Soviet nuclear strike could destroy Paris at any moment, which added considerably to the tension and helped to draw the French and Germans closer.

Not so much the US saying 'rules for thee, but not for me', given their response to 1956 and Soviet nukes in East Germany had been tit for tat. More 'here's the red line' at a time when the US/NATO were on the backfoot, as evidenced by the invasion of Hungary, likely precipitated in part by the Soviets being emboldened after Suez. Another side effect of Suez was the French nuclear weapons programme. They were one of the big losers after Suez. The US nukes that ended up being stationed in Italy/Turkey had orginally been planned to be deployed in France, but France felt it needed its own deterrent after Suez.

Of course, Suez was in turn (partly) caused by the whole Israel switching sides thing. The Soviets ended up supporting the Arab world. Israel became a US ally, where previously they'd had close ties with the Soviet bloc. Czechoslowakian weapons had been used to fight for their independence. There was also obviously the whole Six Day War thing, where the Soviets had supplied to the countries attacking Israel. Luckily that region of the world is no longer part of a cold war between the US, Russia and their respective allies.

TLDR: Not 'rules for thee, but not for me'. More like total chaos, brutal real politik, threats, and red lines as the cold war slowly reached boiling point.