Before revision:  Our second result is a simple parallel batch-dynamic k-clique counting algorithm that uses a newly developed parallel k-clique counting algorithm to bootstrap itself, by enumerating smaller cliques, and intersecting them with the batch. Instantiating this idea gives a simple batch-dynamic algorithm running in O(\Delta(m+\Delta)\alpha^{k-4) expected work and O(\log^{k-2} n) depth w.h.p., all in O(m + \Delta) space.
After revision:  Our second result is ) expected work and O(\log^{k-2} n) depth w.h.p., all in O(m + \Delta) space.