Yes, you often see sources that say a parallax is found from measurements taken 6 months apart. That is rarely the case, and if you think about it, not all stars are visible in the night sky 6 months apart (from the same location on Earth).
In fact a parallax measurement will consist of a series of measurements, possibly taken over more than 1 year. There is no formal requirement for any particular measurement interval, though of course the biggest parallax signal would be if you could get measurements 6 months apart.
In addition to parallax, you need to account for the "proper motion" of the star across the sky. To a first order approximation this is a linear change in right ascension and declination with time (though more complicated models can be used, and you have to account for the ellipticity of the Earth's orbit too). The position of the star in question (with respect to distant background stars) measured in multiple images at multiple times would then be fitted with a function that results in a determination of the parallax and the proper motions in both coordinates. Any motion of the "background stars" is compensated by averaging over all the faint stars in the field. They will all move in a systematic way due to the motion of the Sun in the Galaxy (this is part of the "proper motion"), but their random motions (if they are distant) will be very small and average out well.
A perfect visualisation is provided by HST images of Proxima Centauri which show its path across they sky and illustrates the cyclical motion due to parallax and the linear motion across the sky caused by its proper motion.
The motion of the star towards or away from the Sun could be accounted for from the measurement of the star's line of sight velocity. In practice, these motions are of order tens of km/s and this means it would take tens of thousands of years to change the distance by 1 light year.