  The superconducting diode effect (SDE) is a magneto-electric phenomenon where
an external magnetic field imparts a non-zero center-of-mass momentum to Cooper
pairs, either facilitating or hindering the flow of supercurrent depending on
its direction. We propose that quantum spin Hall insulator (QSHI)-based
Josephson junctions can serve as versatile platforms for non-dissipative
electronics exhibiting the SDE when triggered by a phase bias and an
out-of-plane magnetic field. By computing the contributions from Andreev bound
states and the continuum of quasi-particle states, we provide both numerical
and analytical results scrutinizing various aspects of the SDE, including its
quality Q-factor. The maximum value of the $Q$-factor is found to be universal
at low (zero) temperatures, which ties its origin to underlying topological
properties that are independent of the junction's specific details. As the
magnetic field increases, the SDE diminishes due to the closing of the induced
superconducting gap caused by orbital effects. To observe the SDE, the
QSHI-based Josephson junction must be designed so that its edges are
transport-wise non-equivalent. Additionally, we explore the SDE in a more
exotic yet realistic scenario, where the fermionic ground-state parity of the
Josephson junction remains conserved while driving a current. In this
4$\pi$-periodic situation, we predict an enhancement of the SDE compared to its
2$\pi$-periodic, parity-unconstrained counterpart.
