This kind of question is somewhat tricky when you consider that most modern audiences would be dealing with the Odyssey in some form of translation (which automatically entails at least some degree of alteration on the translator's part). However, at the same time, you must remember that the Odyssey was originally an oral poem, passed down through memorization and meant for recitation (and these origins are reflected in the text as it has come down to us).
Parallelism refers to a specific kind of rhetorical structure, and it can certainly be observed within the Odyssey's first three books. It can even be found in the Odyssey's opening line, in the invocation to the muses (note that, for this answer, I am using the Robert Fagles translation, published by Penguins Classics in 1996):
Sing to me of the man, Muse, the man of twists and turns
This statement that opens the Odyssey is itself an example of parallel structure, where two separate clauses have been balanced against one another. In the very next paragraph, there is a second example of this same grammatical structure:
Many cities of men he saw and learned their minds, many pains he suffered, heartsick on the open sea
Note here the use of repetition. In the first example, the parallelism hinges on the repetition of the word "man" while in the second it hinges upon repetition of the word "many." Note also that both of these examples are taken from the very first paragraphs that open the poem. From this fact alone, we should expect to find examples of parallelism throughout the poem.
In book 2, we find parallelism in Antinous's response against Telemachus and in his complaint against Penelope, which culminates in the following statement:
Great renown she wins for herself, no doubt,
great loss for you in treasure. (transl. Fagles)
In this case, the "great renown" Penelope has garnered for herself is rhetorically balanced against the "great loss" her actions have imposed upon her son.
When reading through these first few books, I would suggest continuing to look for additional examples which reflect a similar structure.