The parallax effect is there,
but it is very small because the stars are so far away that
their parallax can only be observed with very precise instruments.
Indeed, the parallax of stars was not measured conclusively until
the year 1838. Thus, the heliocentric idea of Aristarchus was quickly
forgotten and Western thought stagnated for almost
2000 years as it waited for Copernicus
to revive the heliocentric theory.