As a concept, parallelism means something very similar to what it means in mathematics. Think of parallel lines:
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

They’re straight, they’re equally spaced, and they’re very clearly “parallel.”
Think of the parts of a sentence as lined up, one on top of the next, along their own parallel lines. Consider the sentence “Joe was trying to decide between eating, running, and to walk to the store.” There are three items in the list of activities Joe is considering, so separate these and imagine them on their own parallel lines:

eating
___________________________________________________________
running
___________________________________________________________

to walk
___________________________________________________________
To be parallel, all verbs must look identical. In this case, one sticks out like a sore thumb: “to walk.” Here’s the correct version:

eating
___________________________________________________________
running
___________________________________________________________
walking
____________________________________________________________

https://youtu.be/tXl02zJzNJs

How to recognize parallelism
Parallelism is a rule of English grammar that demands consistency in a sentence’s structure. Any lists of ideas, places, activities, or descriptions that have the same level of importance – whether they are words, phrases, or clauses – must be written in the same grammatical form. Some examples:

activities: running, biking, and hiking
places: the store, the museum, and the restaurant
ideas: how to read, how to write, and how to learn
descriptors: quickly, quietly, and happily

Note the grammatical consistency in each list. The activities all end in “-ing;” the places are all preceded by the article “the;” the ideas all begin with “how to;” the descriptors are all adverbs. In each list, whatever grammatical form is applied to one item is applied to all items. This rule (what applies to one must apply to all) is pretty much all you need to remember.