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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?

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    A) why should there be any relationship between the two concepts? B) it’s very easy to get from C major to A minor, just play a G# followed by A. Or an E major chord followed by A minor chord.
    – Todd Wilcox
    Jan 31, 2019 at 4:36

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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.

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    Well, going C Cm straight usually does not sound that great. More common is to borrow the shared V chord from the minor with natural 7th: C G Cm.
    – yo'
    Mar 13, 2019 at 15:11
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    Whether it sounds great or not is a different matter (and also quite subjective). e.g. I don't think going from C to Cm sounds that bad.
    – nivlac
    Mar 15, 2019 at 2:47
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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.

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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.

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    I've used 'up a semitone' hundreds of times with choirs, and even simple groups of people singing, with not much musical knowledge. Sometimes, because I started in the wrong key, sometimes for fun. Vast majority of the time, no-one seemed to notice, but took the key change in their stride. Helped by the fact that the root of the old key becomes the leading note of the new. Also, with some songs, done a verse or two in parallel minor, when the song is in major. Or vice versa. Again, very few eyebrows raised.
    – Tim
    Jan 31, 2019 at 8:43
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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.

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    But what about when the dominant is not used and it is just going straight from 1 tonic to the parallel minor? Like sometimes I see, especially in Beethoven, C -> Cm and vice versa with no bridging chord at all. I even occasionally see it in Mozart. This gives a weird sounding harmony which is why I consider going to the parallel key to be a modulation, even if the tonic note is shared.
    – Caters
    Jan 31, 2019 at 20:41
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    Two things seem important here. If it's really a move to the parallel key (as in many tangos), there will have to be something that distinguishes the C-minor (for example) from E-major; often that's using a natural B in a dominant-like chord (V,V7,vii0,etc.). Going the other way this isn't so important. The dominant need not be part of the transition but it still connects the keys. The other thing (the obverse of the first) is that one can use "borrowed" chords for color without changing the key (or mode) at all. Besame Mucho is a good example.
    – ttw
    Jan 31, 2019 at 22:21
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Parallel keys are not closely related as they are THREE flats away from each other. However, they can be transitioned to a straightforward and natural way, as they share the same tonic note and have the same dominant chord.

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