Do parsecs scale as an inverse proportion with parallax angle? high school student here. 
In astronomy my teacher says that if the parsec angle ($\alpha$) of a star is 1/50 arcsecond, then that star is 50 parsecs away.
I'm not really convinced - calculation of distances from parsecs requires using the formula $$\text{distance in AU}=\frac{1 AU}{\tan(\alpha)}.$$
This means that distance is inversely proportional to $tan{\alpha}$. Distance is not inversely proportional to $\alpha$.

I know you can use a small angle approximation, but that feels to me as very dodgy - you can get fairly large discrepancies in the calculated distance depending on whether you use $tan(\alpha)$ or $\alpha$ to find the distance.

So, what is the best practice when calculating distances in parsecs? Is my teacher's approximation acceptable in "real life" astronomy?
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)