The first-person camera is positioned behind the center of the player's head, instead of forward (where eyes would be.)
This is very subtle, and it's probably been in Minecraft from the beginning, but it causes some problems with parallax when you tilt your head back or to the side to get a better view around a block.
How to replicate:
Put down 1 stone block, then another on top.
Now put a Brewing Stand on top of that.
Place a half-slab on the ground next to the pillar.
Stand on the half slab, and look straight at the side of the pillar.
(1) Tilt your head back some, so that the player's eyes (at the front of the face) go above the level of the Brewing Stand base, but not so high that you couldn't see the base.
(2) Tilt your head downward some, so your eyes are below the base level, but not so low that you can't see the base.
What I expected:
For (1), in "real life" I'd be able to peer over the top of the Brewing Stand base and see the 3 brewing spots.
For (2), I wouldn't be able to see over the base at all.
What I got:
Switched results. I saw over the Brewing Stand base when looking downward, but not when looking up.
Quick fix:
In (MCP)EntityRenderer.orientCamera(), the part where it positions the first-person camera, change this:
GL11.glTranslatef(0.0F, 0.0F, -0.1F);
to this:
GL11.glTranslatef(0.0F, 0.0F, 0.1F);
The fix doesn't seem to cause problems like seeing through blocks when view bobbing.
(If you're wondering why I'm making an issue of this, it's because my resizing mod makes the POV placement more obvious. The POV felt "wrong" until I shifted the camera forward instead of back.)
Related issues
Comments
The FOV seems fine, even with Quake Pro. I don't get the "seeing through the world" problem when walking close to blocks when view-bobbing.
It may feel like you're "too close" when standing right up against a block if you're not used to it, but if you go into 3rd-person, the block area covered by your head is pretty close to the FOV area in first-person, so it's probably okay.
Is this still a concern in the current Minecraft version? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.

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Reopened due to @unknown's comment in MC-60772.
The changes suggested have been added.
The testing procedure we used involved getting on our knees and try to look at the keyboard on the desk while moving our heads up and down. It has been confirmed the parallax was wrong.
Instead of the suggested 0.1f, we are going for 0.05f, since putting 0.1f move the camera way too close to the blocks at close range. In fact, any value > 0.0f should make the parallax behave properly.
Would it be a sensible suggestion to instead use exactly 0.0? That would fix up a lot of minor FOV problems, including allowing much more accurate creation of title screen panoramas from screenshots, using the FOV slider.
I like your attention to detail!
Does this fix make any problems for the FOV or make it seem like you are "too close" to objects? Those would be my only concerns.