It is a bit harder to explain stellar parallax from the perspective of an Earth-fixed reference frame than from the perspective of a solar system barycentric reference frame. But it’s not all that hard to explain stellar parallax from the perspective of an Earth-centered frame. The motion of the stars can be explained in terms of the “third body effect”. People who model the behavior of satellites in Earth orbit vastly prefer to use an “Earth-centered inertial” reference frame (a non-rotating reference frame with origin at the center of the Earth) than a barycentric frame. From the perspective of such a reference frame, the Sun and Moon (and the planets) make the satellite’s orbit not quite Keplerian. The perturbation is explained by a pseudo-force called the third body effect.
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