Within Europe, they had a massively superior conventional force, to the point that the US almost certainly would have initiated the use of nuclear weapons (tactical or strategic), after being overrun. Part of this is that they really only had to put their forces on their own border, and could support them with lines of communication entirely on land and within their own territory. The US had to pay a lot more to sustain forces in Europe, and keeping the Atlantic open during a major war would have been difficult.
At some point, nuclear forces are pretty much a binary thing -- either you have enough to guarantee a second-strike countervalue capability, or you don't. China seems to believe it can do this mission with ~200 warheads, today.
They actually never had arms parity - USA had over a 20:1 ratio of nukes in the 1960's. The problem is, the :1 is still enough to wreck all sorts of havoc. Also, massively more infantry.
Edit: Sorry, my numbers were wrong, misremebered. 1964: USA 2,416 USSR 375 [1]. So more like 8:1... later on the USSR closed the gap, but the USA had better strategic positions with bases in Taiwan, South Korea, Japan, and West Germany, and they cut off Cuba from the USSR after the Cuban Missile Crisis.
[1] http://en.wikipedia.org/wiki/Nuclear_arms_race