  We consider a space-time fractional parabolic problem. Combining a
sinc-quadrature based method for discretizing the Riesz-Dunford integral with
$hp$-FEM in space yields an exponentially convergent scheme for the initial
boundary value problem with homogeneous right-hand side. For the inhomogeneous
problem, an $hp$-quadrature scheme is implemented. We rigorously prove
exponential convergence with focus on small times $t$, proving robustness with
respect to startup singularities due to data incompatibilities.
