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I got this comment on an answer of mine:

Parallel major/minor keys are not closely related. They are easy to
get to, but not closely related. It's like you might live by a subway
that make it easier to get somewhere, but you may not be close to the
destination.

(A comment on this answer)

Parallel keys are three sharps / flats away from the tonic, and therefore, they are not closely related. Regardless, they are very easy to get to. What's the reason?

The issue here is to define "closely related" and "easy to get to". Here is my take on those phrases:
Closely related: Two keys are closely related when they share similar key signatures.

For example, compared to C major:

• A minor has the same key signature (no sharps/flats)


• F major has only one extra flat


• G major has only one extra sharp

Easy to get to: How smoothly one can transition from one key to another. Most keys can be fairly simple to get to as long as the voice leading is good. Back to our C major example, Playing a C -> Cm chord only requires the E to go to an Eb. So even though C minor has 3 flats, it can be transitioned to in a straightforward way.

Closely related keys are keys that have at most one accidental difference. So the set of notes inside the key are almost identical (or identical in the case of relative major/minor keys).

You can get to any key from any other key, but some key require less perpetration and have easily ways to convincingly transition. Parallel major/minor keys are some of the easiest to transition to and from because they share dominants. A G7 can just as easily get you C or Cm. There's even the concept of a Picardy third which is ending a minor piece or section on a parallel major chord. This is easy to achieve due to how the dominant can go to both naturally and convincingly without any need for preparation.

If you want to see examples on how modulation can take you to any key, I recommend Modulation by Max Reger.

Well, "easy to get to" isn't exactly a very precise term musically. For a good example, note that from A major, B♭ major is almost as unrelated as it gets. However, lots of songs will just shift up a half-step to get to B♭ major.

I suppose the answer is that ultimately, "easy to get to" is completely independent of "related". How related a key is is a good indicator of how easily one can prepare a modulation as smoothly as possible, since we can all agree that "up a half-step" is rarely smooth.

EDIT: I was not aware that "closely related" is a specific term. I'll change it so my original use keeps its meaning.

Parallel keys do share the same dominant; this allows an easy transition using some type of authentic cadence. One may simply have a V-I somewhere (to establish the key) then use V-i to transition. It's not really a modulation; the same tonic note is being used.