Actually you're both right - up to a point.
There is no disputing that the A319 in question, in common with all BA's shorthaul fleet, sends accurate and reliable ADS-B data.
So the trick in interpreting what FR24 tells us here is to believe the data
points (of which there are relatively few covering the landing and taxy) and ignore the
lines that FR24 attempts to join the dots with (because they obviously can't take into account missing plot points).
Looking at the photos, there doesn't seem much doubt that the final position and orientation of the A319 is pretty much as shown on FR24 - roughly parallel to the runway exit, perpendicular to the runway itself, and pointing pretty well directly towards the tower.