YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  f(b()) -> c()
  c() -> c()
  f(c()) -> b()
  a() -> 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:

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

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

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

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