The fact that we see parallax in the distant stars (i.e. that the stars appear to shift by tiny amounts as the Earth moves) is just one of many pieces of experimental evidence that point to the Earth revolving around the Sun. There are many others, like:
- the existence of phases of Venus besides "crescent"
- the fact that we have put vehicles on other planets like the Mars rovers using the heliocentric model
- distance and velocity measurements made by bouncing radar pulses off of planets
- Newtonian gravity being tested in the laboratory and correctly explaining behaviors in other star systems
- and many others
and the fact that they all agree that the Earth revolves around the Sun is why it's a widely-accepted fact. In science, we arrive at the truth by seeing if our explanations for the natural world are consistent with the experiments we do to test them, and if the predictions they make turn out to be accurate. In the case of the heliocentric model, it has satisfied every rigorous test that we put it through* and has predicted the behavior of objects in the Solar System well enough for us to conduct numerous planetary missions, sending scores of probes to orbit or land on the surface of most of the larger astronomical bodies in the Solar System.
*There are numerous YouTube videos out there that claim to debunk the heliocentric model. However, invariably they run into one or more of the following problems: their methodology is not rigorous (i.e. they don't account for sources of uncertainty or bias in their experimental setup), their conclusions aren't supported by the data they take, or they aren't clear enough about their methods for others to repeat and verify their experiment. These problems are why you won't see experiments like this published in peer-reviewed scientific journals (who take the methodology of collecting data and the interpretation of that data very seriously, and tend to reject papers that are not sufficiently rigorous, well-reasoned, and transparent, regardless of the pedigree of their authorship), and they are also often highlighted in the numerous other YouTube channels that have arisen to debunk such claims.