I am looking to find the parent div id (i.e. \"Bakerloo\") from this layout when the button is clicked within \".buttonleft0\" in jquery / javascript.
The parent() function returns the jQuery object, so you need to use the attr() for any of it's attributes like so:
parent()
attr()
$(this).closest().attr('id');
Edit: On further inspection it appears the button isn't the direct child of the div, and so use of the closest() function would be required.
div
closest()