For the Hermitian symmetric $G/P$, Nicolas Perrin explicitly classified all the Schubert varieties admitting a small resolution. (More explicitly, he classifies all the minimal models and quotes a theorem that says a small resolution is a smooth minimal model.) If I remember correctly, except for some very small rank exceptions, type A, and the obvious cases (e.g. projective space), all $G/P$ have a Schubert without a small resolution. Fortunately, there are few enough minimal models for the Hermitian symmetric cases that even in the non-smooth case, their intersection cohomology have a reasonable description, leading to the (previously known) explicit formulas for KL-polynomials. (These formulas are summarized in a paper by Boe that proves the final cases and summarizes and refers to the cases done earlier.)
In type A, there is a Schubert without a small resolution for any 2-step flag variety where the two steps are not adjacent. (The example mentioned in the Zelevinsky paper (which he notes as being due to MacPherson) generalizes to all such cases.) This also means there is a Schubert without a small resolution for any 3-or-more-step flag variety.
This leaves as the only case (barring low rank exceptions) adjacent 2-step flag varieties. I have not seen this resolved in the literature. I suspect there is a Schubert variety in those cases without a small resolution provided the steps are at least two away from both ends. In fact I have an explicit expected counterexample, but I have never learned the tools necessary to prove it actually is one.