YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

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

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

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