Parallel structure has to do with balancing out one's sentence structure. By using words that are similar to each other in a sentence, better sentence fluency and communication are created. For example, the second sentence above is parallel in structure because of the proper use of subject-verb agreement as well as the use of "neither" with the word "nor."
The first sentence might be more easily read as, "In the Grimm Brothers' version, a woodcutter is Red Riding Hood's rescuer who slices open the wolf's belly to let her out." Notice that the word "and" is omitted from the independent clause and the dependent clause to make a more fluid reading of the sentence.
Another way to make the first sentence parallel would be to say, "The woodcutter slices open the worlf's belly to let Red Riding Hood out and rescues her." This version of the sentence takes the two verbs (slices and rescues) and positions them equally in the sentence. Both verbs were necessary to complete the thought of the woodcutter's actions, hence, they are parallel in meaning, structure, and importance. These elements need to be present when constructing parallel sentence structure.