  We consider the classic $k$-center problem in a parallel setting, on the
low-local-space Massively Parallel Computation (MPC) model, with local space
per machine of $\mathcal{O}(n^{\delta})$, where $\delta \in (0,1)$ is an
arbitrary constant. As a central clustering problem, the $k$-center problem has
been studied extensively. Still, until very recently, all parallel MPC
algorithms have been requiring $\Omega(k)$ or even $\Omega(k n^{\delta})$ local
space per machine. While this setting covers the case of small values of $k$,
for a large number of clusters these algorithms require large local memory,
making them poorly scalable. The case of large $k$, $k \ge \Omega(n^{\delta})$,
has been considered recently for the low-local-space MPC model by Bateni et al.
(2021), who gave an $\mathcal{O}(\log \log n)$-round MPC algorithm that
produces $k(1+o(1))$ centers whose cost has multiplicative approximation of
$\mathcal{O}(\log\log\log n)$. In this paper we extend the algorithm of Bateni
et al. and design a low-local-space MPC algorithm that in $\mathcal{O}(\log\log
n)$ rounds returns a clustering with $k(1+o(1))$ clusters that is an
$\mathcal{O}(\log^*n)$-approximation for $k$-center.
