To Him who made great lights, For His mercy endures forever—The sun to rule by day, For His mercy endures forever; The moon and stars to rule by night, For His mercy endures forever.
(Psalms 136:7-9) NKJV
Stellar parallax is used to “verify” that the Earth is moving but it’s dependent upon the A priori assumption that the Earth is moving. In case you weren’t sure, that’s circular reasoning.
Parallax is the apparent change in position of objects due to a change in observation location. You might have noticed this with the iOS background motion. You can also easily see an example of parallax by holding your thumb out in front of your face. Using just your left eye look at where the thumb lines up with some object in the distant background. Now look with just your right eye – the thumb moved, right? That’s parallax since your two eyes are at different locations.
– Wired.com
The measurements are taken 6 months apart from each other when the Earth is on either side of the sun. This is an induction of the premise that the Earth is moving, it’s not a conclusion which has been deduced. The assumption of heliocentricity is implicit within the experimental procedure. This means that stellar parallax can’t be used as proof of heliocentricity.
Here are some examples of the type of induction that’s used to rationalize stellar parallax.

If the earth moves around the sun, then the apparent positions of the stars must change between January and June.

If an astronomer measures the position of a nearby star, and then measures it again six months later…

We only need to look for ways to confirm a theory that has been presented to us.
As with most things, the observations in stellar parallax have two interpretations, one is heliocentric, the other is Geocentrospheric. Your conclusion is dependent on what assumption you started with. The only difference between heliocentric and geocentrospheric theories is relative motion.
Assuming heliocentricity means that parallax distance measurements are based on a triangle which has a base 2 AU (astronomical units) wide. Eliminate this assumption and all of a sudden the stars are not so far away, and therefore, they don’t need to be so massive.