ENH: Support MultiIndex columns in parquet (#34777)

doc/source/whatsnew/v1.2.0.rst

@@ -297,6 +297,7 @@ I/O
 - :meth:`to_csv` did not support zip compression for binary file object not having a filename (:issue: `35058`)
 - :meth:`to_csv` and :meth:`read_csv` did not honor `compression` and `encoding` for path-like objects that are internally converted to file-like objects (:issue:`35677`, :issue:`26124`, and :issue:`32392`)
 - :meth:`to_picke` and :meth:`read_pickle` did not support compression for file-objects (:issue:`26237`, :issue:`29054`, and :issue:`29570`)
+- :meth:`to_parquet` did not support MultiIndex for columns in parquet format (:issue:`34777`)
 
 Plotting
 ^^^^^^^^

pandas/io/parquet.py

@@ -53,9 +53,15 @@ def validate_dataframe(df: DataFrame):
         if not isinstance(df, DataFrame):
             raise ValueError("to_parquet only supports IO with DataFrames")
 
-        # must have value column names (strings only)
-        if df.columns.inferred_type not in {"string", "empty"}:
-            raise ValueError("parquet must have string column names")
+        # must have value column names for all index levels (strings only)
+        if df.columns.nlevels > 1:
+            if not all(
+                x.inferred_type in {"string", "empty"} for x in df.columns.levels
+            ):
+                raise ValueError("parquet must have string column names")
+        else:
+            if df.columns.inferred_type not in {"string", "empty"}:
+                raise ValueError("parquet must have string column names")
 
         # index level names must be strings
         valid_names = all(