Online, I have found many different definitions and comparisons of parataxis and asyndeton. In the dictionary (New Oxford American) they are defined as follows:
Parataxis: the placing of clauses or phrases one after another, without words to indicate coordination or subordination
Asyndeton: the omission or absence of a conjunction between parts of a sentence
I do not see a difference in meaning between these definitions. Curiously, some webpages claim parataxis may include conjunctions.
LiteraryDevices.net describes the devices like this,
Parataxis is derived from a Greek word that means “to place side by side.” It can be defined as a rhetorical term in which phrases and clauses are placed one after another independently, without coordinating or subordinating them through the use of conjunctions. It is also called “additive style.” Parataxis is sometimes used as asyndeton, in which the phrases and clauses are coordinated without conjunctions.
Again, "without coordinating or subordinating them through the use of conjunctions" seems to be equivalent to "coordinated without conjunctions." If, perhaps, there is no coordination regardless of conjunctions in parataxis, how is it possible to distinguish?
What makes "I came, I saw, I conquered" as three independent clauses arranged side by side distinct from reading the statement as "I came, I saw, and I conquered," only with the conjunction removed?
Maybe parataxis can apply to separate sentences as independent clauses, e.g. "I came. I saw. I conquered," while asyndeton is limited to the parts of a single sentence?