diff --git a/fsspec/implementations/cached.py b/fsspec/implementations/cached.py
index 5c495e354..89fab0791 100644
--- a/fsspec/implementations/cached.py
+++ b/fsspec/implementations/cached.py
@@ -355,6 +355,9 @@ def _open(
         self.save_cache()
         return f
 
+    def _parent(self, path):
+        return self.fs._parent(path)
+
     def hash_name(self, path: str, *args: Any) -> str:
         # Kept for backward compatibility with downstream libraries.
         # Ignores extra arguments, previously same_name boolean.
diff --git a/fsspec/implementations/tests/test_cached.py b/fsspec/implementations/tests/test_cached.py
index f47be595a..524e537fe 100644
--- a/fsspec/implementations/tests/test_cached.py
+++ b/fsspec/implementations/tests/test_cached.py
@@ -1271,3 +1271,4 @@ def test_spurious_directory_issue1410(tmpdir):
     # There was a bug reported in issue #1410 in which a directory
     # would be created and the next assertion would fail.
     assert len(os.listdir()) == 1
+    assert fs._parent("/any/path") == "any"  # correct for ZIP, which has no leading /
diff --git a/fsspec/spec.py b/fsspec/spec.py
index c745fdaa3..dffbeb895 100644
--- a/fsspec/spec.py
+++ b/fsspec/spec.py
@@ -887,7 +887,8 @@ def get_file(
             os.makedirs(lpath, exist_ok=True)
             return None
 
-        LocalFileSystem(auto_mkdir=True).makedirs(self._parent(lpath), exist_ok=True)
+        fs = LocalFileSystem(auto_mkdir=True)
+        fs.makedirs(fs._parent(lpath), exist_ok=True)
 
         with self.open(rpath, "rb", **kwargs) as f1:
             if outfile is None: