Perspective confusion in Sky Map mobile app

I have recently ran into an observation which broke my mental schemes, using the Sky Map mobile application.

For an introduction before my actual issue, I should state my understanding that, since the Sun is so far away from us, then all rays casted along any huge area (not involving an actual significant Earth angular change) are almost parallel. Similarly, if looking at the Sun from any point of such area keeping the same head/body rotation, then it should appear basically at the same relative position in our eyes’s field of view. Right?

Now, using this Sky Map app, I had the idea to move the mobile around while keeping its rotation unchanged, and it turns out that the displayed sky scene drastically shifts, literally inducing important eye angle changes for any stars / objets of reference to look at between the initial and final physical body positions. But this shouldn’t make any sense, since these objects appear in reality in the same direction when moving ourselves around without changing angles, like the exposed concept on first paragraph.

How is this even possible?