The first sentence does not have a parallel structure because the first part of the sentence has a subject, a verb, and two adjectives. The second part has only an adjective. (The subject and verb are missing; I don't believe that the fact it has only one adjective is important.) To correct it, you could say:
"Not only was she noisy and rude, but she was also rather stupid.
The second sentence has parallelism, "he was either a liar, or he was a fool." Both parts have a subject, verb and a complement.
The third sentence does not show parallelism. The two parts are almost parallel except that the second part is missing the subject. Here is how to fix it.
"The car is not only economical, but it is also feels good to drive."