diff --git a/jobs/swh-packages.yaml b/jobs/swh-packages.yaml
--- a/jobs/swh-packages.yaml
+++ b/jobs/swh-packages.yaml
@@ -5,7 +5,7 @@
       - "{name}/incoming-tag"
       - "{name}/pypi-upload"
       - "{name}/tests"
-      - "{name}/tox"
+      - "{name}/tox/py3"
       - "debian/packages/{name}"
       - "debian/packages/{name}/update-for-release"
       - "debian/packages/{name}/automatic-backport"
@@ -121,7 +121,7 @@
       - "{name}/incoming-tag"
       - "{name}/pypi-upload"
       - "{name}/tests"
-      - "{name}/tox"
+      - "{name}/tox/py3"
 
 - project:
     name: DSCH
@@ -136,8 +136,8 @@
     pkg: storage
     jobs:
       - 'swh-jobs-{name}'
-      - '{name}/tox':
-          pytest_params: -m cassandra -n 3
+      - '{name}/tox/py3-cassandra':
+          pytest_params: -n 3 # run tests in 3 processes in parallel
 
 - project:
     name: DVAU
diff --git a/jobs/templates/swh-tox.yaml b/jobs/templates/swh-tox.yaml
--- a/jobs/templates/swh-tox.yaml
+++ b/jobs/templates/swh-tox.yaml
@@ -1,5 +1,5 @@
 - job-template:
-    name: '{name}/tox'
+    name: '{name}/tox/{tox_env}'
     display-name: 'Phab. Diff'
     docker-image: swh-tox
     node: "{docker-image}"
@@ -14,6 +14,9 @@
       - string:
           name: DIFF_ID
           description: ID of the Diff patch to apply, if any
+      - string:
+          name: tox_env
+          description: main tox environment to run
       - string:
           name: pytest_params
           description: extra parameters to pass to pytest
@@ -42,7 +45,7 @@
           else
               python3 -m tox -e flake8
           fi
-      - shell: python3 -m tox --result-json tox-results.json -- $pytest_params
+      - shell: python3 -m tox --result-json tox-results.json -e $tox_env -- $pytest_params
 
     publishers:
       - phabricator: