Given from what I know about the USSR, I just don't see how it's on the same level of the USA even at its peak. When I think of a "superpower", I typically think of a country with an enormous ability to project power anywhere around the globe, but the USSR just doesn't seem to have it. Whereas the US was deploying hundreds of thousands of troops overseas into Vietnam with multiple Carrier Battle Groups deployed around the world at once, what exactly could the USSR do? The only main threat it ever seemed to pose to the US was through its nuclear arsenal, but if that made it a superpower then every nuclear-armed country would be one. While it had an immense conventional military, the Soviet army was really only a threat to US allies in Europe (i.e. an actual invasion of the US was out of the question), and I don't know if the Soviet navy or air force had the capability or doctrine to blockade or bomb the US.
Is force projection the main factor in being a superpower? As an example of my thinking, the Falklands War was on the bleeding edge of what the United Kingdom (non superpower) could do in the 80s, with the Vulcan bomber raid against Ascention island being the longest-range bombing run at the time. On the other hand, the US (clear superpower) does multiple similar deployments at the same time as a normal state of operations, not as a heightened state of open warfare with another country.
Political scientists do make a distinction between global power projection and regional power projection. The Soviet Union in my mind is on the border of that — it could project power very extensively on the Eurasian continent, but also had ambitions in Latin America, the Middle East, and Africa, but these tended (outside of Cuba) to be attempts to gather influence rather than project direct military force.
But the term "superpower" to my knowledge has never had a very precise meaning. It essentially relates to the perception that the Cold War revolved around two spheres of influence (the American and the Soviet), and the fact that both of the nations at the center of those spheres had massive militaries and economies was not irrelevant to that. The Soviet Union was a superpower because it was the only bloc of nations that could challenge the other superpower, the United States, essentially. And vice versa, in a somewhat circular definition.
But there has been quite a lot of post-Cold War history that has moved away from seeing the Cold War as being merely about two powers. The role of the Non-Aligned Nations, for example, has been increasing in Cold War scholarship, seeing them not as simply pawns in the superpower race, but as important agents as well. And there has been quite a lot of work about the ways in which those in the US sphere of influence were frequently just as difficult for the US to corral as those outside of it; the French, the West Germans, the British, the Taiwanese, etc., all were more than just vassal states. (I am not sure the same can be said of the non-Soviet members of the Warsaw Pact, but you can say a similar thing about Cuba and North Korea and some of the other states that were not directly subjected to Soviet oversight.)
So the very concept of the superpower being the crucial fulcrum of the Cold War is a somewhat outdated one, even if the concept is not without merit in talking about the spheres of influence (which did exist, even if the nations at their titular centers were not as unduly powerful as the term sometimes implies). It is also entirely valid to point out, as you do, that the two superpowers were not symmetric: the US capability to endanger the Soviet Union, for most of the Cold War, vastly outstripped the capabilities of the Soviets to do the same. In emphasizing a parity, American sources often undercut that fact, to the point that most Americans believe that the US and Soviets were essentially on equal-footing this entire time, which they were not. This is especially the case in nuclear forces; the Soviets did not have a real capability to severely threaten the continental US until well until into the 1960s, and there was no true "parity" in nuclear forces until the mid-1970s.
But to get at your main question: to my knowledge, again, there is no precise definition of a "superpower." Certainly force projection is part of it. Nuclear capabilities are also part of it (and I would caution that there are big differences between a nuclear force projection like the US and Soviets were capable of doing by the mid-to-late Cold War, and those of other nuclear nations — there is a massive gap in nuclear capabilities between those two states and everyone else). And so also are conventional forces. But it is not some simple checklist that one can go down.
A political scientist friend of mine likes to point out that political scientists and historians love appealing to "power" as an explanatory category, but in a vague and totally unquantifiable, unmeasurable way, and I think that particular criticism hits home here. This does not mean the term "superpower" is meaningless, but it is impressionist, not precise.