diff --git a/pandas/io/tests/parser/usecols.py b/pandas/io/tests/parser/usecols.py
index 4fb6ff00e2d7b..c654859f8dc7d 100644
--- a/pandas/io/tests/parser/usecols.py
+++ b/pandas/io/tests/parser/usecols.py
@@ -200,6 +200,26 @@ def test_usecols_with_parse_dates(self):
                            parse_dates=parse_dates)
         tm.assert_frame_equal(df, expected)
 
+        # See gh-13604
+        s = """2008-02-07 09:40,1032.43
+        2008-02-07 09:50,1042.54
+        2008-02-07 10:00,1051.65
+        """
+        parse_dates = [0]
+        names = ['date', 'values']
+        usecols = names[:]
+
+        index = Index([Timestamp('2008-02-07 09:40'),
+                       Timestamp('2008-02-07 09:50'),
+                       Timestamp('2008-02-07 10:00')],
+                      name='date')
+        cols = {'values': [1032.43, 1042.54, 1051.65]}
+        expected = DataFrame(cols, index=index)
+
+        df = self.read_csv(StringIO(s), parse_dates=parse_dates, index_col=0,
+                           usecols=usecols, header=None, names=names)
+        tm.assert_frame_equal(df, expected)
+
         # See gh-14792
         s = """a,b,c,d,e,f,g,h,i,j
         2016/09/21,1,1,2,3,4,5,6,7,8"""