Parallelism means that one or more sentences have been constructed in grammatically similar patterns, and there are several examples of this in The Crisis. With parallelism, the way phrases, clauses, or even sentences are intentionally crafted reflects a balance in the way elements have been written.
Let's first look at a simple example of a sentence (not from The Crisis) that is not written using parallel structure:
I like running and to take pictures.
Here, our direct objects are mismatched. The first one ends in -ing, and the second is an infinitive. We could make the sentence parallel by making both direct objects end in -ing:
I like running and taking pictures.
With that in mind, let's now look for examples of parallel structure in Paine's The Crisis.
The summer soldier and the sunshine patriot will, in this crisis, shrink from the service of his country.
In this example, the sentence begins with dual subjects that reflect parallel structure. Each uses the construction the + adjective + noun.
I call not upon a few, but upon all; not on this State or that State, but on every State.
This sentence utilizes a longer form of parallelism. In fact, there is parallelism within each phrase. Consider the way the first phrase is constructed: "not upon a few, but upon all." There are therefore layers of parallel structure in this construction.
Also consider this example:
The far and the near, the home counties and the back, the rich and the poor, shall suffer or rejoice alike.
This sentence begins with three successive subjects that are constructed using the same pattern: the + noun + and + the + noun.
There are other examples of parallel structure in this piece, so I hope these examples are helpful as you continue your study.