diff --git a/python/cudf/cudf/core/_base_index.py b/python/cudf/cudf/core/_base_index.py
index b29fc475b29..e5945f8860e 100644
--- a/python/cudf/cudf/core/_base_index.py
+++ b/python/cudf/cudf/core/_base_index.py
@@ -342,9 +342,9 @@ def deserialize(cls, header, frames):
     @property
     def names(self):
         """
-        Returns a tuple containing the name of the Index.
+        Returns a FrozenList containing the name of the Index.
         """
-        return (self.name,)
+        return pd.core.indexes.frozen.FrozenList([self.name])
 
     @names.setter
     def names(self, values):
diff --git a/python/dask_cudf/dask_cudf/io/parquet.py b/python/dask_cudf/dask_cudf/io/parquet.py
index ba8b1e89721..810a804e428 100644
--- a/python/dask_cudf/dask_cudf/io/parquet.py
+++ b/python/dask_cudf/dask_cudf/io/parquet.py
@@ -316,7 +316,7 @@ def read_partition(
 
             if index and (index[0] in df.columns):
                 df = df.set_index(index[0])
-            elif index is False and df.index.names != (None,):
+            elif index is False and df.index.names != [None]:
                 # If index=False, we shouldn't have a named index
                 df.reset_index(inplace=True)