I've spent some more time playing with "Upright" and comparing it to DxO's parallel lines feature. Only in very simple circumstances does "Upright" appear to get it right but having said that, equally there are times when DxO's lines feature falls short too.
The problem that the DxO interface has is that it requires me to draw lines on the image. The Lightroom interface doesn't require me to do that but clearly whatever it is using to make guesses about which edges to use is not enough.
Nearly all such images that I have contain visible boundaries where the edges are (contrast, colour change, etc). Rather than draw lines (or have the application guess), I should be able to select an edge as being one that I want to either be horizontal or vertical. I should be able to pick as many such edges across the image as I see fit. Why is this important? Because the lens distortion correction modules in both applications does not always correct the problem so you're still left with curved lines, etc, so the correction required if a line is chosen from the bottom left might be different to the top right.