BUG: ValueError: Given date string not likely a datetime when grouping by PeriodIndex level
- [x] I have checked that this issue has not already been reported.
- [x] I have confirmed this bug exists on the latest version of pandas.
- [x] (optional) I have confirmed this bug exists on the master branch of pandas.

---

#### Code Sample, a copy-pastable example

```pytb
In [10]: t = pd.Series([1], index=pd.PeriodIndex(['2000'], name="A", freq="D"))

In [11]: t.groupby(level="A").size()
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/indexes/period.py in get_loc(self, key, method, tolerance)
    593         try:
--> 594             return self._engine.get_loc(key)
    595         except KeyError:

pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc()

pandas/_libs/index.pyx in pandas._libs.index.IndexEngine.get_loc()

pandas/_libs/index_class_helper.pxi in pandas._libs.index.Int64Engine._check_type()

KeyError: 'A'

During handling of the above exception, another exception occurred:

ValueError                                Traceback (most recent call last)
<ipython-input-11-ca2eac2f77ff> in <module>
----> 1 t.groupby("A").size()

~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/series.py in groupby(self, by, axis, level, as_index, sort, group_keys, squeeze, observed)
   1685             group_keys=group_keys,
   1686             squeeze=squeeze,
-> 1687             observed=observed,
   1688         )
   1689

~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/groupby/groupby.py in __init__(self, obj, keys, axis, level, grouper, exclusions, selection, as_index, sort, group_keys, squeeze, observed, mutated)
    407                 sort=sort,
    408                 observed=observed,
--> 409                 mutated=self.mutated,
    410             )
    411

~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/groupby/grouper.py in get_grouper(obj, key, axis, level, sort, observed, mutated, validate)
    588
    589         elif is_in_axis(gpr):  # df.groupby('name')
--> 590             if gpr in obj:
    591                 if validate:
    592                     obj._check_label_or_level_ambiguity(gpr, axis=axis)

~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/generic.py in __contains__(self, key)
   1848     def __contains__(self, key) -> bool_t:
   1849         """True if the key is in the info axis"""
-> 1850         return key in self._info_axis
   1851
   1852     @property

~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/indexes/period.py in __contains__(self, key)
    384         else:
    385             try:
--> 386                 self.get_loc(key)
    387                 return True
    388             except (TypeError, KeyError):

~/Envs/dask-dev/lib/python3.7/site-packages/pandas/core/indexes/period.py in get_loc(self, key, method, tolerance)
    598
    599             try:
--> 600                 asdt, parsed, reso = parse_time_string(key, self.freq)
    601                 key = asdt
    602             except TypeError:

pandas/_libs/tslibs/parsing.pyx in pandas._libs.tslibs.parsing.parse_time_string()

pandas/_libs/tslibs/parsing.pyx in pandas._libs.tslibs.parsing.parse_datetime_string_with_reso()

ValueError: Given date string not likely a datetime.
```

#### Problem description

With a regular Index, we see the following

```python

In [13]: s = pd.Series([1], index=pd.Index(['a'], name='A'))

In [14]: s.groupby(level="A").size()
Out[14]:
A
a    1
dtype: int64
```

This seems to only affect PeriodIndex. DatetimeIndex works fine.

#### Expected Output

```python
A
2000-01-01    1
Freq: D, dtype: int64
```