I understand that parallax is used to measure distances to stars. But how is the parallax angle actually measured?

In the parallax diagram we have two similar triangles, but we don't know any values other than the distance from one side of the orbit to the other, 2 A.U. Do we know the distance between the background stars? Surely we can't know that information without doing more parallax measurements?

The only explanation I have found is that the telescope knows how many pixels relate to a certain angle, but I want to know how it is measured without this.