We will develop some elements in stochastic analysis in the Wasserstein space $P_2(M)$ over a compact Riemannian manifold $M$, such as intrinsic It\^o formulae, stochastic regular curves and parallel translations along them. We will establish the existence of parallel translations along regular curves, or stochastic regular curves in case where $M$ is a torus. Surprisingly enough, in this last case, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE