Fractional differential equations with delays are ubiquitous in physical systems, a recent example being time-domain impedance boundary conditions in aeroacoustics. This work focuses on the derivation of delay-independent stability conditions by relying on infinite-dimensional realisations of both the delay (transport equation, hyperbolic) and the fractional derivative (diffusive representation, parabolic). The stability of the coupled parabolic-hyperbolic PDE is studied using straightforward energy methods. The main result applies to the vector-valued case. As a numerical illustration, an eigenvalue approach to the stability of fractional delay systems is presented.