YES

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

Consider the left-linear TRS R:

  f(g(x)) -> f(h(x,x))
  g(a()) -> g(g(a()))
  h(a(),a()) -> g(g(a()))

Let C be the following subset of R:

  f(g(x)) -> f(h(x,x))
  g(a()) -> g(g(a()))
  h(a(),a()) -> g(g(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:

  f(g(x)) -> f(h(x,x))
  g(a()) -> g(g(a()))
  h(a(),a()) -> g(g(a()))

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

  phi(f(g(x)) -> f(h(x,x))) = 2
  phi(g(a()) -> g(g(a()))) = 2
  phi(h(a(),a()) -> g(g(a()))) = 1

  psi(f(g(x)) -> f(h(x,x))) = 2
  psi(g(a()) -> g(g(a()))) = 2
  psi(h(a(),a()) -> g(g(a()))) = 1