  Recent experiments reported gate-induced superconductivity in the monolayer
1T$'$-WTe$_2$ which is a two-dimensional topological insulator in its normal
state [1, 2]. The in-plane upper critical field $B_{c2}$ is found to exceed the
conventional Pauli paramagnetic limit $B_p$ by 1-3 times. The enhancement
cannot be explained by conventional spin-orbit coupling which vanishes due to
inversion symmetry. In this work, we unveil some distinctive superconducting
properties of centrosymmetric 1T$'$-WTe$_2$ which arise from the coupling of
spin, momentum and band parity degrees of freedom. As a result of this
spin-orbit-parity coupling: (i) there is a first-order superconductor-metal
transition at $B_{c2}$ much higher than the Pauli paramagnetic limit $B_p$,
(ii) spin-susceptibility is anisotropic with respect to in-plane directions and
results in anisotropic $B_{c2}$ and (iii) the $B_{c2}$ exhibits a strong gate
dependence as the spin-orbit-parity coupling is significant only near the
topological band crossing points. The importance of SOPC on the topologically
nontrivial inter-orbital pairing phase is also discussed. Our theory generally
applies to centrosymmetric materials with topological band inversions.
