I like that the word "parallel"—if written in a sans serif font—contains a parallel. What also contains a parallel, is astronomy and from there, astrology. Keep reading if you want to see me attempt to explain the tilt of the planet.
First of all, let's get one thing in our heads: All the planets, except Pluto, revolve around our sun on the same plane.
Now add to this the fact that the Earth tilts about 23 degrees and this is the reason for seasons. I once really messed explaining this to the point that I called the other person stupid. But I was the stupid one, so let's see if I finally have this straight (hah!) in my head. This is, of course, far easier to understand visually, so grab something and hold it at a tilt and then move it in a circle parallel to the floor, keeping the same tilt, i.e. the object is tilted the same way as you see it all around the circle.
You'll notice that at one point, the bottom is closest to the center of the circle and the top is farthest away. 90 degrees from that—or a quarter of the way around the circle—the whole side of whatever you chose is equidistant to the center of the circle. Another quarter turn and now the top is pointing toward the center of the circle, i.e. closer to it than the bottom is. Yet another quarter turn, and once again the whole side of the object is equidistant to the center of the circle.
I have just described how our Earth looks relative to the sun for, respectively, winter in the northern hemisphere or summer in the southern hemisphere, an equinox, another solstice but with reversed seasons, and another equinox.
The sun in the northern hemisphere climbs very high in the summer time, and above the arctic circle at 66 degrees, 33 minutes north (or 66N33), it is so high, it doesn't set. (The closer you get to the north pole, the more days during the summer you will have this phenomenon.) At the same time, south of the antarctic circle at 66S33, the sun isn't rising at all. And, the closer you get to the south pole, the more days you spend in the winter without sunlight. (These circles are also called polar circles.)
As our planet moves around the sun, it slowly either tilts one way or the other (and here is where I was stupid: The planet doesn't actually move from side to side; it stays fixed in its lean, but appears differently to the sun depending on where in its orbit Earth is). And because of how this tilt changes the angle of sunlight hitting our planet, it seems to us that the sun is climbing higher in the sky as we approach summer (either hemisphere) or lower as we approach winter. That height is called declination. Declination is given as latitude.
So, 00 or zero declination is at the equator. The equinoxes are when the sun is at zero declination (00N or 00S, same thing), and at that moment the sun's rays hit us at a perfect 90 degree angle and day and night are of equal length.
For the summer in the northern hemisphere, the sun climbs northwards to 23N26. (This figure varies slowly over time, but has a range of 22-24 degrees.) This latitude is the maximum northern declination of our sun, and it happens on the summer solstice. Astrologically, that is 00 degrees of Cancer, and the imaginary line at 23 degrees north on planet Earth is called the Tropic of Cancer. It's opposite, the imaginary line that markes the sun's southernmost declinaton is the Tropic of Capricorn. The areas between these two are known as the tropics. The word "tropic" means cycle or turning. We're just drawing huge circles here.
In case you're wondering: 00N declination is 00 Aries, the Spring Equinox and first day of spring in the north. 23N26 is 00 Cancer, the Summer Solstice and first day of summer as well as the year's longest day in the north. As the sun climbs back down to the equator and 00S declination we have the Autumn Equinox at 00 Libra. Finally, the sun goes all the way down to 23S26 and the winter Solstice at 00 Capricorn, giving us northerners our shortest day of the year.
Now that that is clear as mud, what's a parallel?
All the planets do this declination thing. Our tilt along our orbit is also relative to the other planets on the same plane we are. So any planet in a summer sign (between 00 Aries and 00 Libra) will have a northern declination, and any planet in a winter sign (between 00 Libra and 00 Aries) will have a southern declination.
If two planets are at the same declination, let's say they are both at 19N, they are said to be parallel. The symbol for parallel is written as //. If two planets are at the same degree but in opposite declinations, such as one is at 8 degrees north and the other is at 8 degrees south, they are said to be counter-parallel. The symbol for that looks like the // with a single bar across but you can also use the hashtag/pound/flat/octothorpe key: #.
What is this used for in astrology? Parallels have a similar energy to a conjunction (zero degrees apart), i.e. the planets strengthen each other—or crowd each other. Counter-parallels have a similar energy to an opposition (180 degrees apart), meaning they work against each and at best can only take turns being in charge.
One of my interests in this is due to astrological meteorology. Simply put, expect a weather change when the moon changes hemispheres; that is, when the moon crosses the equator or zero degrees declination, i.e. is moving from north to south or vice-versa.
PS: If I totally screwed up the astronomy, PLEASE let me know! Thanks!