They took names and titles from the Romans. They took certain attitudes from the Germans circa WWII via "The Enemy Below".
I agree with R. Star's earlier post about the Trek races not really being anyone in particular. That being said, I think you can read into them what you want, and I've sorta seen the Federation as a whole as the Western powers/NATO, with Earth as America (emerging late into the world/quadrant but quickly becoming the primary power) and the non-human Federation worlds as the various European states (similar in values to Earth/America, but older and more cautious).
In TOS their origins are pretty specific and they played a particular role. They did take on a "life" of their own, though. In past threads on this topic I've put forth the idea that the Vulcans represent the Japanese in some ways. McCoy's line about Vulcan being conquered is part of this.
I wouldn't draw any specific parallers between the various species in ST and the different earth nations. The Federation or at least Earth seems to be somewhat of a socialist utopia. Equality for all, no hunger, no internal conflict* etc. It really doesnt resemble the market economy driven US or any other country of today. (*excluding Section 31 etc.) Klingons seem very much like a medieval society with space travel. Very much of an honor/shame society Perhaps mixed with some European colonialism (British and Dutch Empires etc.). The vulcan aesthetics and some customs definitely seem to be very east Asian (Japanese and Chinese). Look at these captures from ENT episode "Home" and then google traditional Japanese or Chinese home. The influence seems very clear to me.
The parallels aren't really specific. They pretty are broad and general, which is why they are allegorical.