diff --git a/CHANGELOG-unreleased.md b/CHANGELOG-unreleased.md
index 380a8b3c4..e453c5a2c 100644
--- a/CHANGELOG-unreleased.md
+++ b/CHANGELOG-unreleased.md
@@ -13,4 +13,5 @@ the released changes.
 ### Fixed
 - Fixed RTD by specifying theme explicitly.
 - `.value()` now works for pairParameters
+- Setting `model.PARAM1 = model.PARAM2` no longer overrides the name of `PARAM1`
 ### Removed
diff --git a/src/pint/models/timing_model.py b/src/pint/models/timing_model.py
index aa73ca579..65a013d54 100644
--- a/src/pint/models/timing_model.py
+++ b/src/pint/models/timing_model.py
@@ -498,6 +498,16 @@ def __getattr__(self, name):
             f"Attribute {name} not found in TimingModel or any Component"
         )
 
+    def __setattr__(self, name, value):
+        """Mostly this just sets ``self.name = value``.  But in the case where they are both :class:`Parameter` instances
+        with different names, this copies the ``quantity``, ``uncertainty``, ``frozen`` attributes only.
+        """
+        if isinstance(value, (Parameter, prefixParameter)) and name != value.name:
+            for p in ["quantity", "uncertainty", "frozen"]:
+                setattr(getattr(self, name), p, getattr(value, p))
+        else:
+            super().__setattr__(name, value)
+
     @property_exists
     def params_ordered(self):
         """List of all parameter names in this model and all its components.
diff --git a/tests/test_parameters.py b/tests/test_parameters.py
index 3a6a7b832..460bd1ca8 100644
--- a/tests/test_parameters.py
+++ b/tests/test_parameters.py
@@ -605,3 +605,45 @@ def test_correct_number_of_params_and_FD_terms_after_add_or_remove_param():
     m.remove_param("FD4")
     assert len(m.components["FD"].params) == 3
     assert len(m.get_prefix_mapping("FD")) == 3
+
+
+def test_parameter_retains_name_on_set():
+    basic_par_str = """
+    PSR              B1937+21
+    LAMBDA   301.9732445337270
+    BETA      42.2967523367957
+    PMLAMBDA           -0.0175
+    PMBETA             -0.3971
+    PX                  0.1515
+    POSEPOCH        55321.0000
+    F0    641.9282333345536244  1  0.0000000000000132
+    F1     -4.330899370129D-14  1  2.149749089617D-22
+    PEPOCH        55321.000000
+    DM               71.016633
+    UNITS                  TDB
+    """
+
+    m = get_model(io.StringIO(basic_par_str))
+    m.POSEPOCH = m.PEPOCH
+    assert m.POSEPOCH.name == "POSEPOCH"
+
+
+def test_parameter_set_incompatible_fails():
+    basic_par_str = """
+    PSR              B1937+21
+    LAMBDA   301.9732445337270
+    BETA      42.2967523367957
+    PMLAMBDA           -0.0175
+    PMBETA             -0.3971
+    PX                  0.1515
+    POSEPOCH        55321.0000
+    F0    641.9282333345536244  1  0.0000000000000132
+    F1     -4.330899370129D-14  1  2.149749089617D-22
+    PEPOCH        55321.000000
+    DM               71.016633
+    UNITS                  TDB
+    """
+
+    m = get_model(io.StringIO(basic_par_str))
+    with pytest.raises(u.core.UnitConversionError):
+        m.F0 = m.F1