diff --git a/foamlib/_cases/_async.py b/foamlib/_cases/_async.py
index 4113c33..439b36d 100644
--- a/foamlib/_cases/_async.py
+++ b/foamlib/_cases/_async.py
@@ -131,10 +131,13 @@ async def run(
             cmd=cmd, script=script, parallel=parallel, check=check
         ):
             if cpus is None:
-                cpus = 1
-
-                if name == "run" and kwargs.get("parallel", False):
-                    cpus = min(self._nprocessors, cpus)
+                if name == "run":
+                    if kwargs.get("parallel", False):
+                        cpus = max(self._nprocessors, 1)
+                    else:
+                        cpus = 1
+                else:
+                    cpus = 0
 
             async with self._cpus(cpus):
                 await getattr(self, name)(*args, **kwargs)