My Melancholy Blues from Queens’ News of the World.
I’ve always been very partial to Going to California, especially the bridge-like bit that goes :
Seems that the wrath of the Gods
Got a punch on the nose and it started to flow
I think I might be sinking
Throw me a line if I reach it in time
I’ll meet you up there where the path
Runs straight and high
The melody, and Plant’s voice there are haunting.
I’ve recently got into “There’s a Place”, really love it along with the neighbouring “A Taste of Honey” (which is a cover), and have decided to ration my listenings in order to preserve the magic.
There are so many masterpieces on the “A Hard Day’s Night” album, many of which are probably not widely known nowadays. “Things We Said Today” is one of them. The best for me is possibly “I’ll Be Back”, which I can actually listen to repeatedly without it losing the haunting quality that is essential to the Beatles’ genius.
I disagree. “It’s Late” is an all time great rock and roll track from NotW. It’s arguably the best song on the album but way too long to get any airplay.
Both of these feature parallel major/minor changes (like, A major/A minor — this is much rarer than relative minor sequences, like C major/A minor). Perhaps you are drawn to this feature, even if subconsciously.
(“Michelle” and “While My Guitar Gently Weeps” do this, too, but between verses/refrains, not WITHIN a musical phrase like the examples you cited, which is more exciting).
(By the way, I grew up with “I’ll Be Back” as a song in the American LP Beatles ‘65, released about six months after A Hard Day’s Night, so it sounded to me just a little ahead of its time…not quite a bit ahead it’s its time, as it actually was. Ahh, the Sixties, when half a year made such a difference in evaluating musical innovation…)
(Also, I love your mentioning of holding back from listening to a song too often, lest it lose its delicious pungency for you. I don’t recall actively detaining myself in this way, but I know exactly what you’re describing.)
My pick would be “Ripples”. One of my favorite of Hackett’s guitar solos, even when Tony Banks decides to jump in and stomp all over it with his keyboards. (Kind of a musical version of Kanye with his “I’mma let you finish, but…”)
Oh, yeah, sorry. I misused “sleepers” there as meaning songs that don’t get much radioplay, rather than “undiscovered masterpieces.” (For the Pet Sounds discussion, I was using “sleeper” in the latter sense.) I hear all the songs from IV on classic rock radio, but I would agree with you of the lesser-played songs “Going to California” is the best (though I feel it gets reasonable airplay, but I’d be willing to guess it’s the third-least played song on classic radio here off the album.)
I’m also surprised at the mention of “Ramble On.” Great song, one of JPJ’s most iconic bass lines, but I’d guess that’s one of the most played songs on that album (maybe the second most, after “Whole Lotta Love,” though “Heartbreaker/Living Loving Maid” may be in that position), at least with the stations here.
Many, many Beatles songs do it. Beach Boys too. It’s one of the things that makes those groups’ songs so enduring - they threw in subtle chord changes and harmonies that created a push-pull of musical tension or mystery.
Disagree how? No one said there couldn’t be more than one.
Jaycat hit it for six with LLL - one of the best Beatles songs ever.
I don’t hear that in “Eleanor Rigby”. It’s all in E minor (specifically, Dorian). There’s a C major chord in it, but no parallel major E, unless I’m missing it somewhere. “While My Guitar Gently Weeps” does move between A minor and A major, though.
Nobody mentioned Eleanor Rigby.
He might have been thinking of Fixing a Hole (mentioned in OP). McCartney still plays it live and I noticed similar harmony in Say Say Say (with MJ).
I’m having a tough time coming up with many more examples of Beatles songs that have key changes from the major to the parallel minor or minor to parallel major. “Norweigian Wood” fits for major -> parallel minor; “Hey Bulldog” also goes major -> parallel minor. Any others?
ETA: Ah, yes, “Fixing a Hole” fits this shuffling between F major and F minor in the verses.
smack What the hell is wrong with my brain today?
Michelle, Eleanor Rigby, same difference. 
“…that weren’t important yesterday”
So haunting.
I’ll suggest one I’ve mentioned here before: Blues Variation by ELP on the Pictures at an Exhibition album.
No, I’m apparently blind and my brain swaps one Beatles song with a woman’s name as a title with another. Where’s the embarassed emoji? Oh, there actually is one: 
o:o
Supertramp. The album, Even In the Quietest Moments.
The song. Downstream. Such a pure, simple little uncluttered song.
Maybe So What is not a classic album per se, but there was, I think, a fair bit of airplay for its rendition of Turn to Stone, as well as Time Out. Has anyone here ever listened to this song? The bridge alone is breathtaking.
One might say that On Every Street is not a big classic, but a few of its songs got airplay and were covered by others. The song that mostly got ignored, Iron Hand, which can give you chills if you listen to the lyrics (“That night, they, it had even shocked the queen”).
It’s not so much as a “key change” from the major to the parallel minor", so much as simply using the parallel minor as a passing chord to transition to a different chord within the same key They used this technique all the time. They also did the same thing with augmented and diminished chords. They didn’t linger on them, they’d be used strictly as a passing chord, for half a bar or less, but the effect was enough to make even their simplest songs really pack a punch.
Edit - they also did this with the dominant seventh (of a given major chord) a lot. I.e. using it as a passing chord instead of lingering on it.