How does a telescope measure parallax angle?
No, the telescope doesn't measure the parallax. A sextant or any other angle measuring device fit on the telescope does.
And, we don't(can't) directly measure the parallax angle. Instead, we just track the position of the star/object throughout the year. A little bit of spherical astronomy math shows us that the path of a star in the celestial sphere defined by a fixed reference through the year is an ellipse around it's mean postion, called the parallactic ellipse
Here is a link which does the rough math involved.
The semi-major axis of this ellipse is equal to the parllax angle P, while the semi-minor axis is equal to P*sin(b), where b is the stars ecliptic latitude.
EXTRA NOTE: These parallactic ellipses are often rotated and parameters modified as the abberational ellipse is superposed on it. In practice, things get nigh complicated.
In addition: It's often better to measure the angular distance to reference stars, instead of absolute angles, because those angles can be measured with much higher accuracy.
Very true. That is what I meant by "celestial sphere" fixed with some reference(as in the reference stars) :)
@Cheeku As you already indicated in your answer, things get hugely more complex, when improving accuracy or looking to more distant stars in the order of 10s of thousands of lightyears. Everything is in motion to each other, including gravitational aberration, even by bodies within our solar system. The Gaia people are struggling with (and are defining) a reference system taking all those effects into account.