diff --git a/pandas/_libs/tslibs/period.pyx b/pandas/_libs/tslibs/period.pyx
index 878dbcfd99842..a64bf03b5c166 100644
--- a/pandas/_libs/tslibs/period.pyx
+++ b/pandas/_libs/tslibs/period.pyx
@@ -1569,16 +1569,7 @@ cdef class _Period:
             return PyObject_RichCompareBool(self.ordinal, other.ordinal, op)
         elif other is NaT:
             return _nat_scalar_rules[op]
-        # index/series like
-        elif hasattr(other, '_typ'):
-            return NotImplemented
-        else:
-            if op == Py_EQ:
-                return NotImplemented
-            elif op == Py_NE:
-                return NotImplemented
-            raise TypeError(f"Cannot compare type {type(self).__name__} "
-                            f"with type {type(other).__name__}")
+        return NotImplemented
 
     def __hash__(self):
         return hash((self.ordinal, self.freqstr))
@@ -1654,13 +1645,7 @@ cdef class _Period:
                     raise IncompatibleFrequency(msg)
                 # GH 23915 - mul by base freq since __add__ is agnostic of n
                 return (self.ordinal - other.ordinal) * self.freq.base
-            elif getattr(other, '_typ', None) == 'periodindex':
-                # GH#21314 PeriodIndex - Period returns an object-index
-                # of DateOffset objects, for which we cannot use __neg__
-                # directly, so we have to apply it pointwise
-                return other.__sub__(self).map(lambda x: -x)
-            else:  # pragma: no cover
-                return NotImplemented
+            return NotImplemented
         elif is_period_object(other):
             if self is NaT:
                 return NaT
diff --git a/pandas/tests/arithmetic/test_period.py b/pandas/tests/arithmetic/test_period.py
index 0e8bec331ea3f..3b3f5cb90bb31 100644
--- a/pandas/tests/arithmetic/test_period.py
+++ b/pandas/tests/arithmetic/test_period.py
@@ -153,9 +153,11 @@ def test_eq_integer_disallowed(self, other):
         result = idx == other
 
         tm.assert_numpy_array_equal(result, expected)
-        msg = (
-            r"(:?Invalid comparison between dtype=period\[D\] and .*)"
-            r"|(:?Cannot compare type Period with type .*)"
+        msg = "|".join(
+            [
+                "not supported between instances of 'Period' and 'int'",
+                r"Invalid comparison between dtype=period\[D\] and ",
+            ]
         )
         with pytest.raises(TypeError, match=msg):
             idx < other
diff --git a/pandas/tests/scalar/period/test_period.py b/pandas/tests/scalar/period/test_period.py
index 620fc1c006d93..01b61da099481 100644
--- a/pandas/tests/scalar/period/test_period.py
+++ b/pandas/tests/scalar/period/test_period.py
@@ -992,7 +992,8 @@ def test_comparison_invalid_type(self):
         assert not jan == 1
         assert jan != 1
 
-        msg = "Cannot compare type Period with type int"
+        int_or_per = "'(Period|int)'"
+        msg = f"not supported between instances of {int_or_per} and {int_or_per}"
         for left, right in [(jan, 1), (1, jan)]:
 
             with pytest.raises(TypeError, match=msg):