diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 2cadf6ff478..ad57825504a 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -40,6 +40,7 @@ Bug fixes
 
 - Variables which are chunked using dask in larger (but aligned) chunks than the target zarr chunk size
   can now be stored using `to_zarr()` (:pull:`6258`) By `Tobias Kölling <https://github.com/d70-t>`_.
+- Multi-file datasets containing encoded :py:class:`cftime.datetime` objects can be read in parallel again (:issue:`6226`, :pull:`6249`).  By `Martin Bergemann <https://github.com/antarcticrainforest>`_.
 
 Documentation
 ~~~~~~~~~~~~~
diff --git a/xarray/coding/cftimeindex.py b/xarray/coding/cftimeindex.py
index ac6904d4e31..8f9d19d7897 100644
--- a/xarray/coding/cftimeindex.py
+++ b/xarray/coding/cftimeindex.py
@@ -310,7 +310,7 @@ class CFTimeIndex(pd.Index):
     )
     date_type = property(get_date_type)
 
-    def __new__(cls, data, name=None):
+    def __new__(cls, data, name=None, **kwargs):
         assert_all_valid_date_type(data)
         if name is None and hasattr(data, "name"):
             name = data.name
diff --git a/xarray/tests/test_cftimeindex.py b/xarray/tests/test_cftimeindex.py
index 94f0cf4c2a5..c70fd53038b 100644
--- a/xarray/tests/test_cftimeindex.py
+++ b/xarray/tests/test_cftimeindex.py
@@ -1,3 +1,4 @@
+import pickle
 from datetime import timedelta
 from textwrap import dedent
 
@@ -1289,3 +1290,12 @@ def test_infer_freq(freq, calendar):
     indx = xr.cftime_range("2000-01-01", periods=3, freq=freq, calendar=calendar)
     out = xr.infer_freq(indx)
     assert out == freq
+
+
+@requires_cftime
+@pytest.mark.parametrize("calendar", _CFTIME_CALENDARS)
+def test_pickle_cftimeindex(calendar):
+
+    idx = xr.cftime_range("2000-01-01", periods=3, freq="D", calendar=calendar)
+    idx_pkl = pickle.loads(pickle.dumps(idx))
+    assert (idx == idx_pkl).all()
diff --git a/xarray/tests/test_distributed.py b/xarray/tests/test_distributed.py
index f70e1c7958e..a6ea792b5ac 100644
--- a/xarray/tests/test_distributed.py
+++ b/xarray/tests/test_distributed.py
@@ -1,5 +1,8 @@
 """ isort:skip_file """
+import os
 import pickle
+import numpy as np
+import tempfile
 
 import pytest
 
@@ -23,12 +26,15 @@
 
 from . import (
     assert_allclose,
+    assert_identical,
     has_h5netcdf,
     has_netCDF4,
     requires_rasterio,
     has_scipy,
     requires_zarr,
     requires_cfgrib,
+    requires_cftime,
+    requires_netCDF4,
 )
 
 # this is to stop isort throwing errors. May have been easier to just use
@@ -105,6 +111,23 @@ def test_dask_distributed_netcdf_roundtrip(
                 assert_allclose(original, computed)
 
 
+@requires_cftime
+@requires_netCDF4
+def test_open_mfdataset_can_open_files_with_cftime_index():
+    T = xr.cftime_range("20010101", "20010501", calendar="360_day")
+    Lon = np.arange(100)
+    data = np.random.random((T.size, Lon.size))
+    da = xr.DataArray(data, coords={"time": T, "Lon": Lon}, name="test")
+    with cluster() as (s, [a, b]):
+        with Client(s["address"]):
+            with tempfile.TemporaryDirectory() as td:
+                data_file = os.path.join(td, "test.nc")
+                da.to_netcdf(data_file)
+                for parallel in (False, True):
+                    with xr.open_mfdataset(data_file, parallel=parallel) as tf:
+                        assert_identical(tf["test"], da)
+
+
 @pytest.mark.parametrize("engine,nc_format", ENGINES_AND_FORMATS)
 def test_dask_distributed_read_netcdf_integration_test(
     loop, tmp_netcdf_filename, engine, nc_format