Before revision:  Our first result is a new parallel batch-dynamic triangle counting algorithm with O(\Delta\Delta+m) amortized work and O(\log^* (\Delta+m)) depth with high probability (w.h.p.) , and O(\Delta+m) space for a batch of \Delta edge insertions or deletions.
After revision:  Our first result is a new parallel batch-dynamic triangle counting algorithm with O(\Delta\Delta+m) amortized work and O(\log^* (\Delta+m)) depth with high probability , and O(\Delta+m) space for a batch of \Delta edge insertions or deletions.