Does our solar system rotate parallel or perpendicular to the direction of the expansion of the universe? Or is it a little more complicated than that? A friend recently posted a video on Facebook of a 3D model of the solar system rotating like a corkscrew chasing after the sun. It looks completely contrary to everything I can remember the solar system looking like (doesn't the solar system have a plane?) and thought this must be fake. Can anyone shed some light on this? I guess this is really two questions, since the video got me wondering how expansion of the universe relates to the direction of the plane.
Video for your enjoyment?:
https://www.youtube.com/watch?v=0jHsq36_NTU
UPDATE: Okay I found an article that seems to debunk the video quite nicely, which just leaves my initial question about the rotation of the solar system (the plane) in relation to the direction of the expansion of the universe.
Article:
http://www.slate.com/blogs/bad_astronomy/2013/03/04/vortex_motion_viral_video_showing_sun_s_motion_through_galaxy_is_wrong.html
Update 2:
Someone suggested I change the title of this question. That seems to have changed the question entirely -- so I have changed it back. My apologies.
 A: The cosmological principle is that the universe is both homogeneous and isotropic on large scales -the last meaning that it looks the same in every direction.
This principle has been extensively tested and holds up pretty well (there are some debatable anomalies in the cosmic microwave background that are suggested anisotropies, but they are small).
Therefore we must assume that the expansion is isotropic too and therefore whichever way the solar system is oriented - and you can define an axis using the spin axis of the Sun - it has no "special" direction.
Edit: This next part was inserted because the question was edited to ask whether there was alignment of the solar system and galactic disk. It has since been changed back...
That includes within our Galaxy. The "ecliptic plane" is the plane that the planets (approximately) orbit the Sun. As you can see in the pictures below, we are not in a special position with respect to our Milky Way galaxy and neither are the planet orientations anything special. Note that the pictures are not to scale! And that we cannot see our own Galaxy from the "outside"! (Credit: 
http://www.scienceminusdetails.com/2012/06/is-solar-system-tilted-sideways.html )
A nice way to confirm this with your own eyes is to go outside tonight at about 1am (NB this was written 22/11/2014) and look towards the East (assuming the sky is clear and you are in the northern hemisphere). You should be able to spot the planet Jupiter rising and it is between the zodiac constellations of Leo and Cancer. If the solar system plane was lined up with the galactic plane, two things would be true. (i) The planets would always be seen against the background of the Milky Way, which marks the disk of our Galaxy and (ii) the signs of the zodiac, marking the path of the Sun around the sky over the year (the ecliptic) would follow the Milky Way. Neither of these things are true; you should see that the Milky Way rises up into the sky from the south and lies about 50 degrees from Jupiter and well away from Leo and Cancer. (Leo is quite easy to spot).
Top view

Side view (not to scale!) showing orientation of the solar system.

A: 
Does our solar system rotate parallel or perpendicular to the galactic disc?

Neither. You have created a false dilemma. There are lots and lots of other possibilities; in fact, there are an uncountably infinite number of other possibilities. The ecliptic plane is inclined at about a 60 degree angle with respect to the galactic plane. With the possible exception of systems fairly close to the galactic core, the orientation of planetary systems with respect to the galaxy is more or less random.
A: Assuming our solar system is at a 60degree angle in respect to the galactic plane, the planets would still not trail behind the sun like they do in the video. The theory of relativity(as well as our observation of the planets in our solar system) show that the sun's gravity acting on the planets are enough to form a flat plane. If you want to know why things usually flatten out google that, but the short explanation is that no matter what the original conditions are, collisions/gravity will spin the objects into more of a disk formation. This is why we don't see spherical galaxies, and why the majority of our solar system is relatively flat, or on a much smaller scale why the rings around Saturn are so flat. The speed of the solar system going through the galaxy would not cause the planets to trail behind because masses that are in orbit aren't just trailing along holding on by a thread, they are bound rather strongly.
A: The other answers address mainly the solar system's motion relative to our galaxy.
To answer the question in your (time varying) title at the time I saw it:

Does our solar system rotate parallel or perpendicular to the direction of the expansion of the universe?

The answer is neither. There is no "direction" to the universe's expansion. In technical terms, it is simply a real valued, time varying parameter - the so called "scale factor" - in the FLRW metric. 
In practical terms, a lower dimensional mind picture might help you. Imagine a rubber 2-sphere as a model for a two spatial dimension universe, and we draw little pictures on the outside: we'll make them look like galaxies if you like. In particular, draw a straight line segment on the surface. Then we blow the sphere slowly up by pumping air into it. Now, what is the direction of the expansion relative to the line segment's direction? If the expansion is isotropic and the sphere just becomes an every swelling bigger sphere, the proper distances between pairs of points joined by a line segment in any direction increases at the same rate. The expansion is uniform and has no preferred direction. 
A: Another answer to stress that the expansion of the universe 
a) is happening at every point in space by all points receding from all points
b)it is a very weak effective "force", which means it is observable between clusters of galaxies receding from each other. The gravitational energy of galaxies, as well of all their stars and planets allows them to remain compact . At the level of atoms and solid state the electromagnetic forces are even more binding for the stability in size of matter as we know it.
In an analogy of an expanding  raising bread in the oven, the raisins remain intact because the forces holding them together ( electromagnetic) are much stronger then the expanding one pulling them apart. 
Thus there is no direction against which to check the solar system's plane.
