The portion of this statement after a colon is a clause with one subject "it" and a compound predicate of two parts (1) "it is not only..." and (2) "is classified as...." Re-stating "it" is not necessary. "Yet" is a conjunction joining the two parts of the predicate.
However, there is a grammatical error. A series of two parts in a compound predicate are not separated by a comma. Therefore the clause should state, "it is not only completely man-made but also designed exclusively for human use yet is classified as a state park." I admit that a speaker would probably pause before "yet," and that this very long clause needs pauses to be clear. (In this sense, you could argue that the "it" is required because of the comma.)
I believe it would be clearer still if constructed as you suggest: "it is not only completely man-made but also designed exclusively for human use, yet it is classified as a state park."
Notice also that the first verb phrase has compound aspects: "is not only...man-made..but also designed...." This nested compounding is fairly complex and would be clearer if broken up.