YES

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

Consider the left-linear TRS R:

  a() -> c()
  b() -> c()
  f(a(),b()) -> d()
  f(x,c()) -> f(c(),c())
  f(c(),x) -> f(c(),c())
  d() -> f(a(),c())
  d() -> f(c(),b())

Let C be the following subset of R:

  a() -> c()
  b() -> c()
  f(a(),b()) -> d()
  f(x,c()) -> f(c(),c())
  f(c(),x) -> f(c(),c())
  d() -> f(a(),c())
  d() -> 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:

  a() -> c()
  b() -> c()
  f(a(),b()) -> d()
  f(x,c()) -> f(c(),c())
  f(c(),x) -> f(c(),c())
  d() -> f(a(),c())
  d() -> f(c(),b())

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

  phi(a() -> c()) = 4
  phi(b() -> c()) = 3
  phi(f(a(),b()) -> d()) = 6
  phi(f(x,c()) -> f(c(),c())) = 3
  phi(f(c(),x) -> f(c(),c())) = 1
  phi(d() -> f(a(),c())) = 6
  phi(d() -> f(c(),b())) = 6

  psi(a() -> c()) = 5
  psi(b() -> c()) = 5
  psi(f(a(),b()) -> d()) = 7
  psi(f(x,c()) -> f(c(),c())) = 1
  psi(f(c(),x) -> f(c(),c())) = 2
  psi(d() -> f(a(),c())) = 5
  psi(d() -> f(c(),b())) = 5