BUG: Write and read parquet changes index type from PeriodIndex to int values of DataFrame with MultiIndex

[aiwalter]

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• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

# Running example with "DATE" as DatetimeIndex instead PeriodIndex (this works well)
import pandas as pd
data = pd.DataFrame({
  "ID": ["A", "B", "C"],
  "DATE": ["2020-01-01", "2020-01-02", "2020-01-03"],
  "VALUE": [11, 22, 22],
})

data["DATE"] = pd.DatetimeIndex(data["DATE"], freq="D")
types = data.dtypes
data = data.set_index(["ID", "DATE"])
data.to_parquet("data")
assert (pd.read_parquet("data").reset_index().dtypes == types).all()

# NOT running example with "DATE" as PeriodIndex (this fails!)
import pandas as pd
data = pd.DataFrame({
  "ID": ["A", "B", "C"],
  "DATE": ["2020-01-01", "2020-01-02", "2020-01-03"],
  "VALUE": [11, 22, 22],
})

data["DATE"] = pd.DatetimeIndex(data["DATE"], freq="D").to_period()
types = data.dtypes
data = data.set_index(["ID", "DATE"])
data.to_parquet("data")
assert (pd.read_parquet("data").reset_index().dtypes == types).all()

Issue Description

When I write a DataFrame with a MultiIndex that has one level as PeriodIndex, after write and read the PeriodIndex was casted into int values:

Before:

After write and read:

Expected Behavior

it works fine with DatetimeIndex so I would also expect it to work with PeriodIndex

Installed Versions

INSTALLED VERSIONS

commit : 91111fd
python : 3.8.10.final.0
python-bits : 64
OS : Linux
OS-release : 5.4.0-1090-azure
Version : #95~18.04.1-Ubuntu SMP Sun Aug 14 20:09:27 UTC 2022
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : None
LANG : C.UTF-8
LOCALE : en_US.UTF-8

pandas : 1.5.1
numpy : 1.23.4
pytz : 2020.5
dateutil : 2.8.1
setuptools : 52.0.0
pip : 21.0.1
Cython : 0.29.23
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : 2.8.5
jinja2 : 2.11.3
IPython : 7.22.0
pandas_datareader: None
bs4 : None
bottleneck : None
brotli : None
fastparquet : None
fsspec : 0.9.0
gcsfs : None
matplotlib : 3.4.2
numba : None
numexpr : None
odfpy : None
openpyxl : 3.0.7
pandas_gbq : None
pyarrow : 4.0.0
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : 1.6.2
snappy : None
sqlalchemy : None
tables : None
tabulate : 0.8.7
xarray : None
xlrd : None
xlwt : None
zstandard : None
tzdata : None