When is the sum of two (parallel) minimum-phase filters also minimum-phase?
Just looking at it I can see it's related to control theory.  If you make a contrived open-loop gain $$H(z) = \frac{A(z)D(z)}{B(z)C(z)}$$ and then wrap it with unity-gain feedback, you get $$G(z) = \frac{A(z)B(z)C(z)D(z)}{A(z)D(z) + B(z)C(z)}$$.
So any stability test that takes $H(z)$ as an open-loop gain will equally be a test to see if your sum is a minimum-phase system.  If you have some parameter variation then you could use root-locus plots, Bode plot or Nyquist plot analysis, or robust control methods to determine the amount of variation that could be tolerated while keeping the system minimum phase.
As an example, if you let your transfer function be $$k\frac{A(z)}{B(z)} + \frac{C(z)}{D(z)}$$ then you could easily cast this as a root-locus problem, and with a bit of effort you could put it into Bode plot form or feed it into a robust control solution that's designed to work in the $z$ domain.
There's probably a more direct path from here to there than going through control theory, but my hammer says &quot;control theory&quot; on it, and by gosh your problem is looking a lot like a nail to me!