Faulty coordination relates usually to only two connected ideas or examples. There are seven coordinating conjunctions in English, and, so, but, for, yet, nor, and or. They usually appear between two ideas or examples. For instance, look at this sentence.
They wanted to take a long walk, but they didn't like to fight the rain.
The coordinator but connects the two ideas, what they wanted and what they didn't like to do.
Parallelism, on the other hand, can relate to two items but can also relate to many more than two items. More important, parallelism relates to the parts of the sentence or sentences having exactly the same structure. For example, look at this sentence.
They walked the entire street, met several friends, and had a good time.
The structure is parallel because each part has a noun, past tense verb, and direct object.
They walked the entire street,
met several friends,
and had a good time.
If one of the three parts had a different tense or different word order the structure would have a mistake in parallelism.
Practice writing both kinds of writing and you will better understand them.
