The UDHR loves to repeat a couple of phrases. And that's basically what parallel structure is: the repetition of a format.

"Everyone has the right" and "no one shall" recur again and again through most of the 30 articles.

These two formulations both state that something is universal, only from different angles. They're positive rights and negative rights. Positive rights are things that everyone has to have, whereas negative rights are things that you can't do to anyone. The right to social security would be considered a positive right. "No one shall be held in slavery or servitude" (4.1) is an example of a negative right. It's kind of like the yin and yang of human rights philosophy.

Like the list structure, parallelism in the text enhances clarity and makes everything easier to digest. Using the same phrases again and again to define rights removes any confusion over wording and hammers home the points of the declaration.