Having a discrete set of parameters is not enough, as you explain, but having a discrete set of parameters which are continuous rather than semicontinuous in families certainly is. In the case of a reductive group [in characteristic zero] the entire category of representations sheafifies over the spectrum of the center of the enveloping algebra (this is true for any category over the spectrum of its center). In this case we know exactly what this spectrum is (dual Cartan mod Weyl group), and moreover we know the allowable parameters in there form a discrete subset (integral points), and moreover the fiber of the category of group representations over any such parameter is just Vect.
[Edit: this means that the moduli stack of irreducible objects is a disjoint union of $BG_m$'s, coming from the Schur lemma automorphisms.. of course one can twist a family of irreducibles by a line bundle on the base, so the assertion can literally only ever be true up to line bundles. Also I was ignoring connected components of the group, which contribute some additional finite parameters to the spec of the center of the enveloping algebra.]
Edit: to put it slightly more concisely, there are idempotents in the center of the enveloping algebra, when specialized to any G-representation [edit: to be precise, when completed at the augmentation associated to any irrep], which give projections onto integral eigenspaces of the Casimirs. So given a family of irreducibles over a ring you can pick out connected components on which the group acts by a given highest weight representation [edit: up to Schur twist].
[Edit: of course there's a much simpler way to say all this: the values of the Casimirs give locally constant functions on the base of any family of irreducibles, which allow you to pick up the isotypic components on various connected components]