diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index 413dbb9cd0850..f026822bd9d2b 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -454,6 +454,7 @@ Other API changes
 - :meth:`Index.get_indexer_for` no longer accepts keyword arguments (other than 'target'); in the past these would be silently ignored if the index was not unique (:issue:`42310`)
 - Change in the position of the ``min_rows`` argument in :meth:`DataFrame.to_string` due to change in the docstring (:issue:`44304`)
 - Reduction operations for :class:`DataFrame` or :class:`Series` now raising a ``ValueError`` when ``None`` is passed for ``skipna`` (:issue:`44178`)
+- :func:`read_csv` and :func:`read_html` no longer raising an error when one of the header rows consists only of ``Unnamed:`` columns (:issue:`13054`)
 - Changed the ``name`` attribute of several holidays in
   ``USFederalHolidayCalendar`` to match `official federal holiday
   names <https://www.opm.gov/policy-data-oversight/pay-leave/federal-holidays/>`_
diff --git a/pandas/io/parsers/base_parser.py b/pandas/io/parsers/base_parser.py
index 5584730be90e8..b769383281880 100644
--- a/pandas/io/parsers/base_parser.py
+++ b/pandas/io/parsers/base_parser.py
@@ -42,7 +42,6 @@
 from pandas.core.dtypes.cast import astype_nansafe
 from pandas.core.dtypes.common import (
     ensure_object,
-    ensure_str,
     is_bool_dtype,
     is_categorical_dtype,
     is_dict_like,
@@ -395,16 +394,6 @@ def extract(r):
         for single_ic in sorted(ic):
             names.insert(single_ic, single_ic)
 
-        # If we find unnamed columns all in a single
-        # level, then our header was too long.
-        for n in range(len(columns[0])):
-            if all(ensure_str(col[n]) in self.unnamed_cols for col in columns):
-                header = ",".join([str(x) for x in self.header])
-                raise ParserError(
-                    f"Passed header=[{header}] are too many rows "
-                    "for this multi_index of columns"
-                )
-
         # Clean the column names (if we have an index_col).
         if len(ic):
             col_names = [
diff --git a/pandas/tests/io/parser/test_header.py b/pandas/tests/io/parser/test_header.py
index b0742f5b41a92..3fc23525df89e 100644
--- a/pandas/tests/io/parser/test_header.py
+++ b/pandas/tests/io/parser/test_header.py
@@ -557,26 +557,21 @@ def test_multi_index_unnamed(all_parsers, index_col, columns):
     else:
         data = ",".join([""] + (columns or ["", ""])) + "\n,0,1\n0,2,3\n1,4,5\n"
 
+    result = parser.read_csv(StringIO(data), header=header, index_col=index_col)
+    exp_columns = []
+
     if columns is None:
-        msg = (
-            r"Passed header=\[0,1\] are too "
-            r"many rows for this multi_index of columns"
-        )
-        with pytest.raises(ParserError, match=msg):
-            parser.read_csv(StringIO(data), header=header, index_col=index_col)
-    else:
-        result = parser.read_csv(StringIO(data), header=header, index_col=index_col)
-        exp_columns = []
+        columns = ["", "", ""]
 
-        for i, col in enumerate(columns):
-            if not col:  # Unnamed.
-                col = f"Unnamed: {i if index_col is None else i + 1}_level_0"
+    for i, col in enumerate(columns):
+        if not col:  # Unnamed.
+            col = f"Unnamed: {i if index_col is None else i + 1}_level_0"
 
-            exp_columns.append(col)
+        exp_columns.append(col)
 
-        columns = MultiIndex.from_tuples(zip(exp_columns, ["0", "1"]))
-        expected = DataFrame([[2, 3], [4, 5]], columns=columns)
-        tm.assert_frame_equal(result, expected)
+    columns = MultiIndex.from_tuples(zip(exp_columns, ["0", "1"]))
+    expected = DataFrame([[2, 3], [4, 5]], columns=columns)
+    tm.assert_frame_equal(result, expected)
 
 
 @skip_pyarrow
diff --git a/pandas/tests/io/test_html.py b/pandas/tests/io/test_html.py
index d4b78d8371ede..3aac7e95e6591 100644
--- a/pandas/tests/io/test_html.py
+++ b/pandas/tests/io/test_html.py
@@ -14,7 +14,6 @@
 import pytest
 
 from pandas.compat import is_platform_windows
-from pandas.errors import ParserError
 import pandas.util._test_decorators as td
 
 from pandas import (
@@ -918,13 +917,8 @@ def test_wikipedia_states_multiindex(self, datapath):
         assert np.allclose(result.loc["Alaska", ("Total area[2]", "sq mi")], 665384.04)
 
     def test_parser_error_on_empty_header_row(self):
-        msg = (
-            r"Passed header=\[0,1\] are too many "
-            r"rows for this multi_index of columns"
-        )
-        with pytest.raises(ParserError, match=msg):
-            self.read_html(
-                """
+        result = self.read_html(
+            """
                 <table>
                     <thead>
                         <tr><th></th><th></tr>
@@ -935,8 +929,15 @@ def test_parser_error_on_empty_header_row(self):
                     </tbody>
                 </table>
             """,
-                header=[0, 1],
-            )
+            header=[0, 1],
+        )
+        expected = DataFrame(
+            [["a", "b"]],
+            columns=MultiIndex.from_tuples(
+                [("Unnamed: 0_level_0", "A"), ("Unnamed: 1_level_0", "B")]
+            ),
+        )
+        tm.assert_frame_equal(result[0], expected)
 
     def test_decimal_rows(self):
         # GH 12907