By parallel, I assume you mean on the same plane. You are kind of right, on diagrams they are exactly on the same plane because well you are looking at a 2D representation. In reality, they are not exactly on the same plane. That said, there are pretty close to it, compared to the Sun's equator, Mercury is only at 3 degrees of inclination, same with Venus, the Earth is at 7 degrees, Mars and Saturn at 5, Jupiter, Uranus and Neptune at 6. So they are all pretty close to be perfectly on the same plane. Why?

Well first of all the solar system started as a big cloud of particles. All of them were moving around in random direction. You know a figure skating spins? Well it show phenomenon, if you spin and you bring back your limb toward the center of your body, you will spin faster. Same thing happened to the solar system. As gravity brought the cloud of particle together, it started to spin faster.

Particles started to hit each other, which created a favored direction of spin. Aka most stuff was spinning in one direction, so everything else go bumped into, leaving just one main direction of spinning. You can see that happening in this demonstration video at around 2:30min.

Now you have a ball of particle spinning in one direction. The second analogy I will use is when you are spinning a pizza dough. I won't go into the physics detail of why this is happening, but basically the fact that the dough is spinning will stretch it around the axis of rotation. Same happened with the cloud of particles that was our early solar system. It stretch around the axis of rotation, bringing all the particles into roughly the same plane where all the particles eventually came together with gravity into the different planets.