diff --git a/fsspec/implementations/reference.py b/fsspec/implementations/reference.py
index d5d3f2968..3d22ddc40 100644
--- a/fsspec/implementations/reference.py
+++ b/fsspec/implementations/reference.py
@@ -41,7 +41,7 @@ def _first(d):
 
 def _prot_in_references(path, references):
     ref = references.get(path)
-    if isinstance(ref, (list, tuple)):
+    if isinstance(ref, (list, tuple)) and isinstance(ref[0], str):
         return split_protocol(ref[0])[0] if ref[0] else ref[0]
 
 
@@ -173,8 +173,11 @@ def open_refs(field, record):
             """cached parquet file loader"""
             path = self.url.format(field=field, record=record)
             data = io.BytesIO(self.fs.cat_file(path))
-            df = self.pd.read_parquet(data, engine=self.engine)
-            refs = {c: df[c].to_numpy() for c in df.columns}
+            try:
+                df = self.pd.read_parquet(data, engine=self.engine)
+                refs = {c: df[c].to_numpy() for c in df.columns}
+            except IOError:
+                refs = None
             return refs
 
         self.open_refs = open_refs
@@ -871,6 +874,9 @@ def cat(self, path, recursive=False, on_error="raise", **kwargs):
                 # found and on_error is "raise"
                 try:
                     u, s, e = self._cat_common(p)
+                    if not isinstance(u, (bytes, str)):
+                        # nan/None from parquet
+                        continue
                 except FileNotFoundError as err:
                     if on_error == "raise":
                         raise
diff --git a/fsspec/implementations/tests/test_reference.py b/fsspec/implementations/tests/test_reference.py
index d82dc1771..fefdd6024 100644
--- a/fsspec/implementations/tests/test_reference.py
+++ b/fsspec/implementations/tests/test_reference.py
@@ -792,3 +792,44 @@ def test_deep_parq(m, engine):
         "instant/one/.zarray",
         "instant/one/0",
     ]
+
+
+def test_parquet_no_data(m):
+    zarr = pytest.importorskip("zarr")
+    lz = fsspec.implementations.reference.LazyReferenceMapper.create(
+        "memory://out.parq", fs=m
+    )
+
+    g = zarr.open_group(lz, mode="w")
+    arr = g.create_dataset(
+        name="one",
+        dtype="int32",
+        shape=(10,),
+        chunks=(5,),
+        compression=None,
+        fill_value=1,
+    )
+    lz.flush()
+
+    assert (arr[:] == 1).all()
+
+
+def test_parquet_no_references(m):
+    zarr = pytest.importorskip("zarr")
+    lz = fsspec.implementations.reference.LazyReferenceMapper.create(
+        "memory://out.parq", fs=m
+    )
+
+    g = zarr.open_group(lz, mode="w")
+    arr = g.create_dataset(
+        name="one",
+        dtype="int32",
+        shape=(),
+        chunks=(),
+        compression=None,
+        fill_value=1,
+    )
+    lz.flush()
+    arr[...]
+
+    assert arr[...].tolist() == 1  #  scalar, equal to fill value
diff --git a/fsspec/mapping.py b/fsspec/mapping.py
index 8fb9b9efb..752eef352 100644
--- a/fsspec/mapping.py
+++ b/fsspec/mapping.py
@@ -112,7 +112,7 @@ def getitems(self, keys, on_error="raise"):
             for k, v in out.items()
         }
         return {
-            key: out[k2]
+            key: out[k2] if on_error == "raise" else out.get(k2, KeyError(k2))
             for key, k2 in zip(keys, keys2)
             if on_error == "return" or not isinstance(out[k2], BaseException)
         }