YES

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  h(f(f(f(h(c(),h(b(),c()))))),h(f(f(b())),b())) -> f(h(b(),f(a())))
  f(c()) -> b()

Let C be the following subset of R:

  h(f(f(f(h(c(),h(b(),c()))))),h(f(f(b())),b())) -> f(h(b(),f(a())))
  f(c()) -> b()

The parallel critical pair system PCPS(R,C) is:

  (empty)

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  h(f(f(f(h(c(),h(b(),c()))))),h(f(f(b())),b())) -> f(h(b(),f(a())))
  f(c()) -> b()

All parallel critical peaks (except C's) are decreasing wrt rule labeling:

  phi(h(f(f(f(h(c(),h(b(),c()))))),h(f(f(b())),b())) -> f(h(b(),f(a())))) = 1
  phi(f(c()) -> b()) = 1

  psi(h(f(f(f(h(c(),h(b(),c()))))),h(f(f(b())),b())) -> f(h(b(),f(a())))) = 1
  psi(f(c()) -> b()) = 1