Certain comparison operations misbehaving for period dtype
This is closely related to https://github.com/pandas-dev/pandas/issues/28930.

```python
import pandas as pd

s = pd.Series([pd.Period("2019"), pd.Period("2020")], dtype="period[A-DEC]") 
s == "a"                                                                                                                                                     
```

yields

```
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-4-3a654234a428> in <module>
----> 1 s == "a"

~/pandas/pandas/core/ops/__init__.py in wrapper(self, other)
    527         rvalues = extract_array(other, extract_numpy=True)
    528 
--> 529         res_values = comparison_op(lvalues, rvalues, op)
    530 
    531         return _construct_result(self, res_values, index=self.index, name=res_name)

~/pandas/pandas/core/ops/array_ops.py in comparison_op(left, right, op)
    253 
    254     if should_extension_dispatch(lvalues, rvalues):
--> 255         res_values = dispatch_to_extension_op(op, lvalues, rvalues)
    256 
    257     elif is_scalar(rvalues) and isna(rvalues):

~/pandas/pandas/core/ops/dispatch.py in dispatch_to_extension_op(op, left, right, keep_null_freq)
    134 
    135     try:
--> 136         res_values = op(left, right)
    137     except NullFrequencyError:
    138         # DatetimeIndex and TimedeltaIndex with freq == None raise ValueError

~/pandas/pandas/core/arrays/period.py in wrapper(self, other)
     98             result.fill(nat_result)
     99         else:
--> 100             other = Period(other, freq=self.freq)
    101             result = ordinal_op(other.ordinal)
    102 

~/pandas/pandas/_libs/tslibs/period.pyx in pandas._libs.tslibs.period.Period.__new__()
   2459                 value = str(value)
   2460             value = value.upper()
-> 2461             dt, _, reso = parse_time_string(value, freq)
   2462             if dt is NaT:
   2463                 ordinal = NPY_NAT

~/pandas/pandas/_libs/tslibs/parsing.pyx in pandas._libs.tslibs.parsing.parse_time_string()
    265             yearfirst = get_option("display.date_yearfirst")
    266 
--> 267     res = parse_datetime_string_with_reso(arg, freq=freq,
    268                                           dayfirst=dayfirst,
    269                                           yearfirst=yearfirst)

~/pandas/pandas/_libs/tslibs/parsing.pyx in pandas._libs.tslibs.parsing.parse_datetime_string_with_reso()
    291 
    292     if not _does_string_look_like_datetime(date_string):
--> 293         raise ValueError('Given date string not likely a datetime.')
    294 
    295     parsed, reso = _parse_delimited_date(date_string, dayfirst)

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

but the output should be `Series([False, False])`.  The other comparison operators `<`, `<=`, `>`, and `>=` yield the same error, which also seems incorrect (the error message should probably be different, and the error should be a `TypeError` rather than a `ValueError`).  This is happening on 0.25.1 and master.