Seems to be reasonable as lomg as the datum from the initial reading can be maintained - one measurement would be 6 months seperate from the second. So whatever the first angular pposition was made on/with/against would have to be maintained until 6 months later.
Quick read of Earthsky say the distance to the nearest is 300,000 time that of the distance to the sun.
So the parallax would be the angle that 2/300,000 so 1/15000 degrees.
so angle = tan-1(1/150,000)
Quick run through a calculator and you get a parallax of 1.375 arcsec.
Which seems within the performance of present equipment.
Unsure of an easy way to do it. One that comes to mind is center a more distant star that is in line with our nearest and measure the angular separation, wait 6 months, repeat measurement. Nearest star should have changed position by 1.375 arc min, further star should not have moved owing to the increased distance and it's hopefully negliable parallax.
As in most of this expect to get several measurements, then I guess discard the lowest and highest and average the remainder. Could also discard the 2 highest and 2 lowest, might tighten up the final "average" measurement. Maybe take 20 measurements and discard the highest 10% and lowest 10% is probably what I would consider, so from 20 only 16 go forward.
Will depend on the number of measurements taken and how good you are. If the measurements are a general mess and all over the place then something has gone wrong.