It is certainly possible to enable or disable the home site's navigation on your sub-site's navigation. This is done through the Site Admin Properties screen. You can access this by:
1) Going to the sub-site > click toolbox > Site Admin > Properties (from the right nav).
2) Going to Admin > Assets > select your Sub-site > click 'Properties' (from the right nav).
The setting will be called "pin home site navigation in left column".
Check or uncheck this setting and click 'Save'.