diff --git a/pandas/tests/series/methods/test_fillna.py b/pandas/tests/series/methods/test_fillna.py
index 47703b294c2b1..9fded26c37caf 100644
--- a/pandas/tests/series/methods/test_fillna.py
+++ b/pandas/tests/series/methods/test_fillna.py
@@ -21,6 +21,7 @@
     isna,
 )
 import pandas._testing as tm
+from pandas.core.arrays import period_array
 
 
 class TestSeriesFillNA:
@@ -946,3 +947,26 @@ def test_datetime64tz_fillna_round_issue(self):
         )
 
         tm.assert_series_equal(filled, expected)
+
+    def test_fillna_parr(self):
+        # GH-24537
+        dti = date_range(
+            Timestamp.max - Timedelta(nanoseconds=10), periods=5, freq="ns"
+        )
+        ser = Series(dti.to_period("ns"))
+        ser[2] = NaT
+        arr = period_array(
+            [
+                Timestamp("2262-04-11 23:47:16.854775797"),
+                Timestamp("2262-04-11 23:47:16.854775798"),
+                Timestamp("2262-04-11 23:47:16.854775798"),
+                Timestamp("2262-04-11 23:47:16.854775800"),
+                Timestamp("2262-04-11 23:47:16.854775801"),
+            ],
+            freq="ns",
+        )
+        expected = Series(arr)
+
+        filled = ser.fillna(method="pad")
+
+        tm.assert_series_equal(filled, expected)