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.