  A gauge PDE is a natural notion which arises by abstracting what physicists
call a local gauge field theory defined in terms of BV-BRST differential (not
necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular
attention to globally well-defined definitions and equivalences of such
objects. We demonstrate that a natural geometrical language to work with gauge
PDEs is that of $Q$-bundles. In particular, we demonstrate that any gauge PDE
can be embedded into a super-jet bundle of the $Q$-bundle. This gives a
globally well-defined version of the so-called parent formulation. In the case
of reparameterization-invariant systems, the parent formulation takes the form
of an AKSZ-type sigma model with an infinite-dimensional target space.
