Learn English – “Still” versus “Yet”

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 example:

Those changes need to be made but the plan to make those changes does not yet exist.

is grammatically correct. The implication here is that the plan is expected to exist at some point in the future, but currently does not exist -- perhaps it is the next item on the work schedule.

Your second example:

The plan to make those changes has not yet existed.

sounds quite strained. It seems almost like a line from The Hitchhiker's Guide To The Galaxy or any other text dealing with time-travel, and would not find a home in natural English conversation.

If you wished to make use of still, you could indeed say:

The plan to make those changes still does not exist

or, phrased more naturally:

We still don't have a plan as to how to make those changes

In either of these two cases, as @PeterShor notes, the implication is that you expected the plan to be ready by this point, but for some reason it is not.