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