YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

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

  phi(f(a()) -> b()) = 3
  phi(f(a()) -> f(c())) = 3
  phi(a() -> d()) = 3
  phi(f(d()) -> b()) = 2
  phi(f(c()) -> b()) = 1
  phi(d() -> c()) = 2

  psi(f(a()) -> b()) = 3
  psi(f(a()) -> f(c())) = 3
  psi(a() -> d()) = 3
  psi(f(d()) -> b()) = 2
  psi(f(c()) -> b()) = 1
  psi(d() -> c()) = 2