“19) Tycho Brahe famously argued against the heliocentric theory in his time, positing that if the Earth revolved around the Sun, the change in relative position of the stars after 6 months orbital motion could not fail to be seen. He argued that the stars should seem to separate as we approach and come together as we recede. In actual fact, however, after 190,000,000 miles of supposed orbit around the Sun, not a single inch of parallax can be detected in the stars, proving we have not moved at all.”
The closest star system to ball Earth is Alpha Centauri. It is 4.367 light years away (2.572 × 1013 miles). That is 25,720,000,000 miles. If you draw Earth’s orbit (distance from sun is about 150,000,000 miles) and these two triangles it would look like this (green is summer and orange is winter):
Now in this scenario, an observer might be able to measure a difference in parallax. However, this is the best case scenario with the closest star assumed to be perpendicular to the orbit of Earth.
There is no reason to expect the stars to seem to move. They do of course, but because of the distances, the visible change is minimal. It is possible to measure this difference, and it has been performed since 1672.
Additionally, the stars do change since the night sky is different depending on the season. Here are pictures of the how the big dipper and other constellations change by location and with different times of the year.