Since these bits of debris crossing Earth's orbit have the same relative speed, why aren't they drawing parallel paths in the upper atmosphere? (consider one hour of observation to minimise Earth's rotation influence)
Meteoroids belonging to the same stream (so, this is valid not only for the Perseids stream) actually do follow the same trajectory (orbit) in space but meteors caused by them apparently appear to originate from a particular spot in the sky (the so called radiant) trailing in all possible directions away from it only due to our perspective of view. In fact, you could even spot a so called stationary meteor which would appear point-like because it appears exactly in the radiant and is directed along your line of sight.
Usually, to explain this it is compared to the same effect when you'd drive a car at a high velocity through a snow-fall: the snowflakes would apparently appear only to you to come from some frontal area radiating then in all directions away from it, although they all have the same trajectory in space for an external non-moving observer.
They are all following parallel paths. They are not on exactly the same trajectory but they are all travelling parallel to each to other in their orbit around the sun, in a stream of particles.
What you see is then multiple meteors on parallel paths. When you look in the direction that the meteors are coming from you see them appearing to radiate from a point
These meteors are all travelling parallel to each other. It is only perspective that makes them appear to be radiating from a point.
This is similar to the "flying through space" screensaver that used to be popular on windows.
Why don't all perseid meteors follow the same trajectory?
Since these debris crossing Earth's orbit have the same relative speed, why aren't they drawing parallel paths in the upper atmosphere ?
When a comet travels through space it disintegrates leaving a shower of debris, every year the Earth passes through the tail's remnants (between July 14 - August 24 this year). The comet nucleus is 26 km in diameter, thus it produces a wide tail of debris that can fall to Earth from various angles, appearing in the direction of the constellation Perseus (but not actually coming from that constellation).
The meteor shower appears to radiate from a single point.
That is a result of passing through the trail of debris.
See NASA's webpage: "Perseids":
"Where Do Meteors Come From?
Meteors come from leftover comet particles and bits from broken asteroids. When comets come around the sun, they leave a dusty trail behind them. Every year Earth passes through these debris trails, which allows the bits to collide with our atmosphere and disintegrate to create fiery and colorful streaks in the sky.
Their radiant — the point in the sky from which the Perseids appear to come from — is the constellation Perseus. This is also where we get the name for the shower: Perseids. However, the constellation for which a meteor shower is named only serves to aid viewers in determining which shower they are viewing on a given night. The constellation is not the source of the meteors.
The answer has been given, it is the effect called "perspective." I would simply add an example that should make the situation quite clear: sunbeams. When you see sunbeams come through clouds, it is natural to think they are not parallel rays, because they look very far from parallel. Your eye tells you that the Sun is sending out those rays in all directions, from its apparent location in the center of the clouds. But your eye cannot tell that the Sun is a trillion times farther than the clouds, so it cannot tell the divergence of the rays is a complete illusion. Actually those rays are almost perfectly parallel, because they all point to that incredibly distant Sun. But look at them carefully the next time you see sunbeams through the clouds-- they seem as far from parallel as they could possibly be. So it is with the streaks of meteors, it's because your eye cannot pick out that third dimension so tends to imagine it is not important.