We present a massively parallel algorithm, with near-linear memory per machine, that computes a $(2+\varepsilon)$-approximation of minimum-weight vertex cover in $O(\log\log d)$ rounds, where $d$ is the average degree of the input graph. Our result fills the key remaining gap in the state-of-the-art MPC algorithms for vertex cover and matching problems; two classic optimization problems, which are duals of each other. Concretely, a recent line of work---by Czumaj et al. [STOC'18], Ghaffari et al. [PODC'18], Assadi et al. [SODA'19], and Gamlath et al. [PODC'19]---provides $O(\log\log n)$ time algorithms for $(1+\varepsilon)$-approximate maximum weight matching as well as for $(2+\varepsilon)$-approximate minimum cardinality vertex cover. However, the latter algorithm does not work for the general weighted case of vertex cover, for which the best known algorithm remained at $O(\log n)$ time complexity.