  For distributions over discrete product spaces $\prod_{i=1}^n \Omega_i'$,
Glauber dynamics is a Markov chain that at each step, resamples a random
coordinate conditioned on the other coordinates. We show that $k$-Glauber
dynamics, which resamples a random subset of $k$ coordinates, mixes $k$ times
faster in $\chi^2$-divergence, and assuming approximate tensorization of
entropy, mixes $k$ times faster in KL-divergence. We apply this to obtain
parallel algorithms in two settings: (1) For the Ising model
$\mu_{J,h}(x)\propto \exp(\frac1 2\left\langle x,Jx \right\rangle + \langle
h,x\rangle)$ with $\|J\|<1-c$ (the regime where fast mixing is known), we show
that we can implement each step of $\widetilde \Theta(n/\|J\|_F)$-Glauber
dynamics efficiently with a parallel algorithm, resulting in a parallel
algorithm with running time $\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$.
(2) For the mixed $p$-spin model at high enough temperature, we show that with
high probability we can implement each step of $\widetilde \Theta(\sqrt
n)$-Glauber dynamics efficiently and obtain running time $\widetilde O(\sqrt
n)$.
