The thing I don't get is why do we need to sweep out at least half a circle and then look at the angular difference. Why can't we just measure the angle at the maximum point of parallax? It seems better than having to half a year just measure the same angle twice. So why is it that we wait 6 months to take 2 readings?
As you know that Parallax angles of less than 0.01 arc sec are very difficult to measure from Earth because of the effects of the Earth's atmosphere.
Generally the parallax shift for stars are very very small. thus we have to wait for six months for earth to rotate a significant distance in its orbit allow us to have the parallax shift in our measurable range.
The larger the base ($BC$ vs. $EF$), the smaller angles of the rays from the base, the greater angle between rays (closer to the right angle), and - consequently - the more precise measuring.
Compare: Can you tell, where is the intersection of the blue rays?
Both pairs of rays have their intersection points in the same level (hight). But the red rays ($h$ and $i$) allow us to determine the point of intersection more accurate.
Astronomical distances are large. So, a 'one parsec' distant object has one second of parallax for a full Earth-orbit displacement. That's the equivalent of 5 parts per million of a radian, about the same angle as a dime seen face-on at a distance of a mile.
It isn't easy to accurately measure those small angles, so any result that one can get at one week, is less accurate than (and will be superseded by) a half-year observation.