YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

  phi(h(a(),a()) -> f(c())) = 1
  phi(a() -> a()) = 1
  phi(a() -> a()) = 0
  phi(b() -> h(c(),a())) = 1

  psi(h(a(),a()) -> f(c())) = 1
  psi(a() -> a()) = 1
  psi(a() -> a()) = 0
  psi(b() -> h(c(),a())) = 1