diff --git a/qutebrowser/browser/downloads.py b/qutebrowser/browser/downloads.py
index 28f20b0ef..fbbf4ff11 100644
--- a/qutebrowser/browser/downloads.py
+++ b/qutebrowser/browser/downloads.py
@@ -1264,7 +1264,7 @@ class DownloadModel(QAbstractListModel):
         else:
             return ""
 
-    def data(self, index, role):
+    def data(self, index: QModelIndex, role: int = Qt.ItemDataRole.DisplayRole) -> Any:
         """Download data from DownloadManager."""
         if not index.isValid():
             return None
diff --git a/qutebrowser/browser/network/proxy.py b/qutebrowser/browser/network/proxy.py
index 62872d68e..5b29c29fc 100644
--- a/qutebrowser/browser/network/proxy.py
+++ b/qutebrowser/browser/network/proxy.py
@@ -4,10 +4,10 @@
 
 """Handling of proxies."""
 
-from typing import Optional
+from typing import Optional, List
 
 from qutebrowser.qt.core import QUrl, pyqtSlot
-from qutebrowser.qt.network import QNetworkProxy, QNetworkProxyFactory
+from qutebrowser.qt.network import QNetworkProxy, QNetworkProxyFactory, QNetworkProxyQuery
 
 from qutebrowser.config import config, configtypes
 from qutebrowser.utils import message, usertypes, urlutils, utils, qtutils
@@ -71,7 +71,7 @@ class ProxyFactory(QNetworkProxyFactory):
             capabilities &= ~lookup_cap
         proxy.setCapabilities(capabilities)
 
-    def queryProxy(self, query):
+    def queryProxy(self, query: QNetworkProxyQuery = QNetworkProxyQuery()) -> List[QNetworkProxy]:
         """Get the QNetworkProxies for a query.
 
         Args:
diff --git a/qutebrowser/completion/models/completionmodel.py b/qutebrowser/completion/models/completionmodel.py
index 6ddf27dcf..808608e85 100644
--- a/qutebrowser/completion/models/completionmodel.py
+++ b/qutebrowser/completion/models/completionmodel.py
@@ -4,9 +4,9 @@
 
 """A model that proxies access to one or more completion categories."""
 
-from typing import MutableSequence
+from typing import MutableSequence, overload, Optional, Any
 
-from qutebrowser.qt.core import Qt, QModelIndex, QAbstractItemModel
+from qutebrowser.qt.core import Qt, QModelIndex, QAbstractItemModel, QObject
 
 from qutebrowser.utils import log, qtutils, utils
 from qutebrowser.api import cmdutils
@@ -30,7 +30,7 @@ class CompletionModel(QAbstractItemModel):
         self.column_widths = column_widths
         self._categories: MutableSequence[QAbstractItemModel] = []
 
-    def _cat_from_idx(self, index):
+    def _cat_from_idx(self, index: QModelIndex):
         """Return the category pointed to by the given index.
 
         Args:
@@ -48,7 +48,7 @@ class CompletionModel(QAbstractItemModel):
         """Add a completion category to the model."""
         self._categories.append(cat)
 
-    def data(self, index, role=Qt.ItemDataRole.DisplayRole):
+    def data(self, index: QModelIndex, role: int = Qt.ItemDataRole.DisplayRole) -> Any:
         """Return the item data for index.
 
         Override QAbstractItemModel::data.
@@ -74,7 +74,7 @@ class CompletionModel(QAbstractItemModel):
         idx = cat.index(index.row(), index.column())
         return cat.data(idx)
 
-    def flags(self, index):
+    def flags(self, index: QModelIndex) -> Qt.ItemFlag:
         """Return the item flags for index.
 
         Override QAbstractItemModel::flags.
@@ -91,7 +91,7 @@ class CompletionModel(QAbstractItemModel):
             # category
             return Qt.ItemFlag.NoItemFlags
 
-    def index(self, row, col, parent=QModelIndex()):
+    def index(self, row: int, col: int, parent: QModelIndex = QModelIndex()) -> QModelIndex:
         """Get an index into the model.
 
         Override QAbstractItemModel::index.
@@ -108,7 +108,15 @@ class CompletionModel(QAbstractItemModel):
             return self.createIndex(row, col, self._categories[parent.row()])
         return self.createIndex(row, col, None)
 
-    def parent(self, index):
+    @overload
+    def parent(self, index: QModelIndex) -> QModelIndex:
+        ...
+
+    @overload
+    def parent(self) -> Optional[QObject]:
+        ...
+
+    def parent(self, index=None):
         """Get an index to the parent of the given index.
 
         Override QAbstractItemModel::parent.
@@ -116,6 +124,9 @@ class CompletionModel(QAbstractItemModel):
         Args:
             index: The QModelIndex to get the parent index for.
         """
+        if not index:
+            return QObject.parent(self)
+
         parent_cat = index.internalPointer()
         if not parent_cat:
             # categories have no parent
diff --git a/qutebrowser/utils/objreg.py b/qutebrowser/utils/objreg.py
index c0715d90a..8a3489d09 100644
--- a/qutebrowser/utils/objreg.py
+++ b/qutebrowser/utils/objreg.py
@@ -77,7 +77,7 @@ class ObjectRegistry(collections.UserDict):  # type: ignore[type-arg]
 
         super().__setitem__(name, obj)
 
-    def __delitem__(self, name: str) -> None:
+    def __delitem__(self, name: _IndexType) -> None:
         """Extend __delitem__ to disconnect the destroyed signal."""
         self._disconnect_destroyed(name)
         super().__delitem__(name)
@@ -101,7 +101,7 @@ class ObjectRegistry(collections.UserDict):  # type: ignore[type-arg]
                 pass
             del partial_objs[name]
 
-    def on_destroyed(self, name: str) -> None:
+    def on_destroyed(self, name: _IndexType) -> None:
         """Schedule removing of a destroyed QObject.
 
         We don't remove the destroyed object immediately because it might still
@@ -111,7 +111,7 @@ class ObjectRegistry(collections.UserDict):  # type: ignore[type-arg]
         log.destroy.debug("schedule removal: {}".format(name))
         QTimer.singleShot(0, functools.partial(self._on_destroyed, name))
 
-    def _on_destroyed(self, name: str) -> None:
+    def _on_destroyed(self, name: _IndexType) -> None:
         """Remove a destroyed QObject."""
         log.destroy.debug("removed: {}".format(name))
         if not hasattr(self, 'data'):