The parallax from Earth is insignifigant in regards to casual viewing, but can be detected withmore sofisticated instruments. There is also some small “waggle” in the Moon’s rotation, so that you can see a very small amount of the far side from time to time. Also, the distance to the moon varies enough to affect the amount of light seen during a solar eclipse; I don’t know if this has any effect on the lit portion of the moon, though.
The phase of the moon depends on the orbit of the moon around the earth, which is ~28 days. The part of the moon that is lit by the sun changes cyclicly at about that rate. Where you are on the earth doesn’t change what part of the moon is lit by the sun, it only changes what part of the moon you can see. But the moon is far enough away from the earth that there is hardly any difference looking at the moon from one side of the earth vs. the other. Just a sliver of difference.
Because of different time zones, people on opposite sides of the earth are likely to be looking at the moon 12hrs apart, thus about 1/56 of the lunar orbit, so there will be a slight difference in observed phase on the “same” night.
A crescent moon can certainly be oriented in different ways compared to the horizon. This could be different at different parts of the earth at the same time, not because a different part of the moon is lit, but because the ‘sky’ and the ‘earth’ are in different places relative to the moon.
The phases are noticeably different in the two hemispheres. When I’ve been in the northern hemisphere I’ve noted, for example, that the waxing crescent moon is reversed in shape i.e. it’s the right side of the moon that is lit, rather than the more normal left side.
The opposite side of the moon will seem to be lit in Australia and England- in the northern hemisphere, a waxing crescent moon is D-shaped (the right side is lit), while a waning crescent moon is C-shaped (the left side is lit). It’s the other way around in the southern hemisphere.
When I went to Australia last November, I noticed that the moon was upside down from what I’m used to. If any of you Sydney Dopers saw a tourist walking around the harbor with her head tipped way over, that was me. I wasn’t fully convinced that the moon was upside down (it just looked different, and it was hard to say how) until I tipped my head to see.
I thought about it a while and realized that, on the equator, the downward-facing side of the crescent moon must be lit. It would be tipped slightly one way or the other at different times of year.
It would do that (and you’re much more observant than most of my non-major astronomy students ever were). The way I think about this uses astrological signs.
Disclaimers: I don’t believe in astrology. I know that the signs don’t correspond with the constellations. This just gives me some handy even divisions of the ecliptic to work with, and means I don’t have to talk about right ascension or any of that. I know about the effect of precession on the equinoxes and solstices, but I am ignoring it here to keep things simple.
Say it’s spring (the sun is in Aries) and the waxing crescent moon is 60 degrees, or two signs, ahead of it in Gemini. The ecliptic in Gemini is north of where it is in Aries (where it crosses the celestial equator). The lighted side of the moon will face toward the sun, so the lighted side of this crescent moon will point west and south.
Now say it’s summer (sun in Cancer, moon in Virgo). The ecliptic in Cancer is north of where it is in Virgo- the northern summer solstice is in Cancer. The lighted side of this crescent moon will point west and north.
Now say it’s fall (sun in Libra, moon in Sagittarius). The ecliptic in Sagittarius is south of where it is in Libra, so the lighted side of this crescent moon will point west and north.
Now say it’s winter (sun in Capricorn, moon in Pisces). The ecliptic in Capricorn is south of where it is in Pisces- the northern winter solstice is in Capricorn. The lighted side of this crescent moon will point west and south.
A similar analysis applies for waning crescent moons, but the moon is two signs behind the sun, so in spring and winter it points east and north, in summer and fall it points east and south.
When the sun is in Gemini or Sagittarius, the waxing crescent moon will point directly west. When the sun is in Leo or Aquarius, the waning crescent moon will point directly west.
I had a long long long drunken argument with a friend in a pub over this. I had just come back from the Maldives and mentioned how odd it was to see a crescent moon with the points upwards. He could not get his head round this at all because
He finally got it when sober though. I think we broke £60 on the bar bill, which may account for his delayed comprehension.