YES

# parallel critical pair closing 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:

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

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R follows from that of C.

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  f(x,c()) -> f(c(),c())
  f(c(),x) -> f(c(),c())

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

  f(x,c()) -> f(c(),c())
  f(c(),x) -> f(c(),c())

Let C be the following subset of R:

  (empty)

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

# emptiness

The empty TRS is confluent.