Do parsecs scale as an inverse proportion with parallax angle?
For angles of magnitude 1/50 arcsecond (of the order $10^{-7}$, the difference between $\tan\alpha$ and $\alpha$ is truly negligible (if alpha is in radians) and only a matter of a choice of units if you don't. It is quite correct to say: distance (in AU) = 1/angle (in radians)  to get the distance from the angle in radians.

Using parsecs a means that we don't need to use a different unit for the angle, and the point of parsecs is that we can use "arcseconds".
So it is correct to use distance (in parsec) = 1/parallax angle (in seconds)