So find the shortest PA+BQ.

There are three situations.

On the left or right, it is easy to forget that the length is longer than the intersection, so it is excluded.

When they intersect, in fact, the length of PA+BQ is the same, as long as pq and ab intersect.

Maybe you were misled by PC = 8 in your analysis. When they intersect, PC is always 8.

The picture I drew extends backwards, and then I can draw a parallelogram, PC = AB.

In the analysis, saying that PC is equal to 8 is equivalent to saying that pq and ab intersect.

As I said before, it's longer on the left or right than the intersection.

So they are the shortest when they intersect, that is, the shortest when PC = 8.