Blender - Inherit parent null name to children meshes

import bpy

# Collect mesh objects that have an EMPTY (null) as their direct parent
parent_to_meshes = {}

for obj in bpy.data.objects:
    if obj.type == "MESH" and obj.parent and obj.parent.type == "EMPTY":
        parent = obj.parent
        parent_to_meshes.setdefault(parent, []).append(obj)

# Rename meshes based on their direct parent's name
for parent, meshes in parent_to_meshes.items():
    # Sort by original name to ensure predictable numbering
    meshes.sort(key=lambda o: o.name)

    for index, mesh_obj in enumerate(meshes):
        if index == 0:
            mesh_obj.name = parent.name
        else:
            mesh_obj.name = f"{parent.name}_{index}"