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
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