Correct usage of "parallel" versus "in parallel" versus "parallelly"

It is a question of grammar. You're looking for a word to qualify the way A and B are used, in the construct “to verb A and B qualifier”. This qualifier has to be an adverb (“a word or phrase that modifies or qualifies an adjective, verb, or other adverb or a word-group”).

Now, you want to express parallelism. “Parallel” itself is an adjective, as well as a noun and a verb. It is not an adverb, and as such, cannot be used in “using A and B parallel”. The natural adverb that derives from parallel is parallelly; though it's not exactly very common in general usage, it does exist and is attested in multiple (though not all) dictionaries. So, “using A and B parallelly” works.

Regarding in parallel, it so happens that it is a common phrase meaning “occurring at the same time and having some connection”. It may be more commonly used than parallelly, which is why it would feel very natural in your sentence, but both are correct.

Your three examples are not sentences, so I will extend them with "They will be ...":

"They will be using A and B parallel" is wrong (unless there is something called "B parallel") as parallel is an adjective or noun but probably needs to be an adverb in this position. "They will be using A and B parallel to each other" looks slightly better.

The other two are strictly correct as adverbs, though care is needed to avoid using parallelly when an adjective is needed. The much more common phrase in parallel, usually following what it describes, can be used either as a adjectival phrase or as an adverbial phrase. My guess is that in parallel is more popular in part simply because it is less likely to be used wrongly.